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ADVANCED 


ALGEBRA 


BY 


WM.    T.    WELCKER 

Graduate  of  West  Point, 

PROFESSOR   OF  MATHEMATICS  IN  THE  UNIVERSITY  OF  CALIFORNIA, 

AUTHOR    O^ 

"  WELCKER'S  MILITARY  LESSONS,"  ."  WELCKER'S  PRIMARY 
ARITHMETIC"    (unpublished),     "WELCKER'S  PRAC- 
TICAL ARITHMETIC"  (unpublished),   Etc. 


SAN  FRANCISCO: 

W.  M.  HINTON  &  CO.,  Book  and  Job  Printers 

536  Clay  Street. 


Entered  according  to  Act  of  Congress  in  the  year  1880,  by  William  Thomas  VVelcker,  in 
the  office  of  the  Librarian  of  Congress  at  Washington, 


PREFACE. 


This  small  volume  contains  what  remains  of  the  course 
in  Algebra,  after  matriculation,  to  the  students  in  the 
Colleges  of  Civil  Engineering,  Mines,  and  Mechanic  Arts 
in  the  University  of  California. 

It  is  intended  as  a  continuation  of  the  excellent  work 
on  algebra  by  Mr.  John  B.  Clarke,  of  the  Mathematical 
Department  of  the  University;  and  it  is  thought  it  will, 
in  connection  with  Clarke's  Algebra,  or  with  any  work 
of  similar  scope,  furnish  a  good  and  sufficient  preparation 
for  those  who  intend  to  pursue  the  higher  mathematics. 

The  constant  aim  and  endeavor  throughout  has  been  so 
to  present  the  various  topics  discussed  as  to  render  them 
easy  of  comprehension  by  the  undergraduate  student. 


WM.  T.  WELCKER. 


Berkeley,  Caxifornia, 
July,  1880. 


J0514(> 


CONTENTS 


ARTICLE.  PAGE. 

Summatiou  of  series  of  fractious  of  certain  forms 2-5  1-5 

Method  of  De  Moivre   6-7  5-10 

If  a  is  a  root,  first  member  divisible  by  a  and  conversely.  9  11 

Elimination 10-14     12-17 

Every  equation  has  at  least  one  root 17  18 

Every  equation  of  the  inth  degree  has  m  roots 18  19 

Composition  of  Equations 20  20 

Every  equation  of  an  odd  degree  has  at  least  one  real  root  28  24 
Every  equation  of  even  degree  and  absolute  term  negative 

has  two  real  roots  of  different  signs 29  25 

Every  equation  has  an  even  number  of  real  positive  roots 
when  absolute  term  positive;  an  odd  number  if  term 

is  negative 30  25 

Changing  signs  of  alternate  terms  changes  signs  of  roots  31  26 

Des  Cartes'  Kule " 32  27 

De  Gua's  Criterion 34  29 

To  transform  to  an  equation  whose  roots  shall  be  multi- 
ples of  former  roots 36  30 

To  clear  of  fractions  and  yet  keep  coefficient  of  highest 

power  unity 37  30 

To  transform  to  an  equation  whose  roots  shall  be  recip- 
rocals of  the  former  roots 39  33 

Eecurring  Equations 42-43  34 

To  transform  to  an  equation  whose  roots  shall  be  squares 

of  the  former  roots 44  34 

To  transform  to  an   equation  where  the  roots  shall  be 

greater  or  less  by  a  certain  quantitj' 45  35 

Another  mode  of  finding  transformed  coefficients 46  37 

Synthetical  Division 47  38 

To  transform  to  an  equation  wanting  a  second  or  any 

other  particular  term 49  41 

Derived  Polynomials 50  43 


VI  '  CONTENTS. 

ARTICLE.  PAGE. 

Relations   of  Derived   Polynomials   to   the  roots  of  an 

equation 51  45 

To  discover  equal  roots,  if  any 52-53  47-48 

Kational  Integral  Function .  53  50 

Any  term  of  such  function  can  be  made  to  contain  the 

sum  of  all  which  precede  or  succeed 54  50-51 

Law  of  Continuity 56  51 

Limits  of  Roots,  definitions  of 57  52-53 

MacLaurin's  Limit 53 

Ordinary  Superior  Limit  of  Positive  Roots 59  54-55 

Inferior  Limit  of  Positive  Roots 60  55-56 

Superior  Limit  of  Negative  Roots 61  56 

Inferior  Limit  of  Negative  Roots 62  56 

Newton's  Limit 63  56 

Budan's  Test  of  Imaginary  Roots 64  58 

If  the  substitution  of  p  and  q  give  different  signs  there  is 

one  real  root  between  them 65  61 

When  there  may  be   an   odd  number  of  roots  between 

them  and  when  an  even  number 66  61 

Sturm's  Theorem.     Its  Object 67  62 

Enunciation  of  Sturm's  Theorem 68  63 

Three  Lemmas 69-71  64 

Demonstration  of  Sturm's  Theorem 72-74  65-71 

Cardan's  Solution  of  Cubic  Equations 86  73 

Solution  of  Equations  of  the  4th  Degree 88  77 

Waring's  Method 89  77 

Waring's  Method  applicable  only  to  equations  having  two, 

and  but  two,  imaginary  roots 90  78 

Occasional  solution  of  higher  equations 91  80 

Cubic  equations  having  special  relations  between  the  co- 
efficients    92  81 

Detection  of  whole  number  roots 93  83 

Solution  of  recurring  equations 97  86 

Recurring  equations  of  higher  even  degrees  solved  by  one 

of  a  degree  half  as  high 98  87 

Exponental  Equations 99  90 

Approximate  solutions  of  higher  numerical  equations. .  .  100  90 

Horner's  Method 101  90 

Demonstration  of  Horner's  Method 102  91 

Manner  of  shortening  the  calculations 103  95 

Newton's  Method 105  100 

Fourier's  Conditions 106  101 

Trigonometric  solution  of  cubic  equations 107  103 


PpCIPLES  OF  ALGEBilA, 

CHAPTEE    I. 

Summation   of  Series. 

Art.  1.  In  addition  to  the  treatment  of  this  subject  in 
Chirk's  Algebra,  a  little  more  will  be  here  added.  It  was 
there  shown  that  when  a  series  w^as  so  constituted  that 

each  term  might  be  derived  from  the  expression      J^ 

by  giving  a  constant  value  to  i^  and  suitable  values  to  q 
and  k,  the  sum  of  n  terms  of  the  series  could  be  found  by 

forming  two  auxiliary  series  from  the  expressions^   and 

~  —  respectively,  subtracting  n  terms  of  the  second  from 
k-\7p       r  J 

the  corresponding  n  terms  of  the  first,  and  then  dividing 

the  difference  by  I?. 

Art.    2.      If    d     series    of   fractions     have    the    form 
its  sum  is  equal  to  the  difference  between  a  se- 


k{k-\ij){L-VAp) 

ries  whose  terms  Jiare  the  form    — ,        .   and   another    whose 

terms  J  tare  the  form  77— — ,/,    ,  ^  ,   divided  by  2p. 
For 

q       _  q  _  q{k±p){k-i-2p)-qk(k-j-p)  _ 

kik-yp)      (k-\-p){k-]-2p)  k{k-fp)\k-\~2p)  ~ 

-k(k-^py)ik-^2p)        =  kik.^pm-2pY  -^^  *^^^  ^'^'^''^ 
by  2p  gives  the  form  proposed.     If  any  term  of  the  pro- 


Z  PRINCIPLES  OF  ALGEBRA. 

posed  can  be  found  in  tliis  way,  the  sum  of  n  terms  of  that 
series  may  be  had  by  taking  the  difference  between  the 
sums  of  the  n  corresponding  terms  of  the  two  auxihary  se- 
ries and  dividing  it  by  2p. 

Ex.  1.     What  is  the  sum  of  the  series 

3  9         J^ 21  ^ 

5.8.11  +  8.11.14  +  11  AAAl  +  14.17.20  ^  ^^^  ' 

Comparing  with  the  formula  above,  we  see  that  (/^=  3, 
9,  15,  etc.;  p  =  d;  and  k=5,  8,  11,  14,  etc.  Therefore 
the  auxiliary  series  are: 

^'^^    k{k-\-p)'         5T8  +  8.11^ii:i4  -^'    •     '    •    ' 


From 


3(2?t— 3)     •    .         3(2/1—1) 
+  (3n— 1)  (3n+2)  +  (3?i+2)(3n-L5) 

3  9 


(yfc+p)(yt+2j9)  •        8.11   '   11.14     '     '    '    *    ' 

3(2n— 3)  3(2?i— 1) 


(3?i+2)(3H  +  5)  ^  (3?H-5)(3?7-f8) 


and  the  sum : 


3  3(2»— 1)      -I  ^      1 


A_r — _    3(2»— 1)   1 

5 .  3L  5 . 8 "~  (3n4-5)(3n+8) J 

to  infinity. 


2.315.8      (3n-f5)(3n-f8)J    '     8.11^11.14 
1 


14.17 

Now  the  sum  of  this  last  series  to  infinity  r=  -^.    Hence 

the  sum  of  n  terms  of  the  proposed  series  is: 
lr£_         271—1         -]        1 
2140      (3H-f5)(3?i+8)J  +24' 

and  when  n  =  oc, 

r  2       11 


1      1  n       n' 

2 


40~   /„    ,    5\/      ,   8 


(3  +  .)(^+'). 


J.  _  1       1        13^ 

+"24"8U+24~  240' 


SUMMATION  OF  SERIES.  ii 

The  sum  of  n  terms  of  Example  1  = 

Irl  2n—l         ^^  1,1, 

-   77^  —  ,^     ,  ^,,» — r-^v    -\~ 4-  -,  1   -.T  4-  .  .  to  n — 1  terms 

2L40      (3n4-5)(3?x+8)J   '^8.11^11.14^ 

Ex.  2.     What  is  the  sum  of  an  infinite  number  of  terms 

Of  the  series  172^  +  ^^  +  37475  +  etc  ? 

Ans.  IJ 

1  4 

Ex.  3.     What  is  the  sum  of  the  series  -|-  ^-^-^  4-   ' 

1.0.0      0.0.7 

5T7T9  +  779711  +  ^''•'  *°  '"''"'y  •  '^'''-  24- 

Art.    3.     If   a   series  of  fractions  have  the  terms  of  the 

form   Yjj- — r/i—rrr-^jT-n—s  >  its  sum  will  be  equal  to  the 

difference  between  the  sums  of  two  series  whose  terms  have 
respectively  the  forms 

g  and  ^ 

/ciki-p)(k^2p)  {k+p)(k-i-2p){k-{-dp) 

divided  by  3p.  In  these  formulas  p  is  constant,  and  q  and 
k  have  suitable  values  assigned  to  them.  The  proof  of  the 
above  is  easily  seen  by  performing  the  subtraction: 


k{k+p)(k^2p)        {k-{-p){kJr2p)(k^3p) " 
(k+p)(k-\r2p)(qk^dpq)  -  (k^p)(kJr2p)qk 
k{kJrPnk+2py{k+Sp) 

3pg 

k(k+p)(kJr2p)(k^dpy 


Ex.  4.     What  is  the  sum  of  n  terms  and  of  an  infinite 

Qb 

8, 


g2  rj.2 

number  of  terms  of   the  series      -   ^  ^    .      -[-    ^^    .   ^  -|- 

1.2.3.4      '      2.3.4.5    ' 


3.4.5.6 


+    etc? 


PRINCIPLES  OF  ALGEBRA. 


Here  g --=  6^  T,  8^   9^  etc.;  p  =  l;   k=l,  2,  8,  4,  5, 
etc.     Hence  the  auxiliary  series  are: 


2.3.4"^  3.4.5"^     '     ■     '    '    n(n+l)(/i+2)   '^ 


(n+l)(7i4-2)(n  +  3) 


3L1.2.3      0i+l)(n+2)(7i+3)J"^3\2.3.4^ 


and  /S'  =^  „  .  ^    ^  „ 

(n+l)(n+2)(n+3) 

(0:5  +  4:576  ^^'^*"^^^| 


The  sum  of  this  last  series  to  infinity  is,  by  preceding^ 


methods  ^^T^; 


1 


6  — 


1/13  15  17  ,      ,        i  \ 

+  3  (2X4  +  3.1:5  +  4.5.(5  "■  ^*<'-'  **'  "  *^™^^-) 

Whenn  =  oo,S=l(6  +  ;-^)  =  |. 
Ex.  5.     AVhat  is  the  sum  of  the  series 

i:ir5  .T + 375^ +5.7 .1,11'  "^  ^"'^'^''y  ■     ^"'-  7^2  ■ 

Art.  4.  A  consideration  of  the  laws  of  the  preceding 
series  and  the  mode  of  their  summation  will  show  that  the 
sum  of  any  series  of  fractions  of  the  form 

k{k+p){/c+2pY .     .     .     .    ik-i-mp) 

is  equal  to  —    of    the    difference   between   two    series  of 
mp 

fractions  of  the  forms,  respectively, 

and 


k{k-^p)    .     .     [^f(m— l)/jj  ik-^p)(k-\-2p)  .   .   .  [k~\  mp 


StiMMATlON  OF  SERIES.  5 

Art.   5.     If  a  series  of  fractions  has  the  terms  as  fellow: 

k'^'kik^py'  k{k^--p){kyip)'^  '  '  ~^k{k^p)  .  .  .  (k~{-mp) 

(in  which  the  last  term  is  a  general  term),  its  sum  will  be 
equal  to  the  difierence  of  two  series  of  fractions,  having  re* 
spectiveh^  the  forms: 

e(c4-/>)   .   .   .   (c^-mp)  r(c'4-p)   .   .   .    [c-[-(m+l)y>] 

k(kJrp)  .   .-.   [k^(m-l)p]  ^"^         k{k-\-p)   .   .   .  (k^mp) 
divided  by  k — c — j),  as  will  be  seen  by  performing  the  in- 
dicated subtraction. 

2      2  4 

Ex.  1.     Find  the  sum  of  vi  terms  of  the  series  ._  -[-  ^^  -|- 

2.4.6 

ir5r7  +  «'*'- 

Here  p  =  2,  c  =  2  and  k==S,  and  the  auxiliary  series  are : 
,.   ,   2.4  ^   2.4.6  2.4.6   ....  [2-i-2m— 2]       , 

^^-^+-3X+     •    •    •    3Xr .    .       .l3+2,^-4]""^ 
^4_^  2.4.6  2.4.6  ....  2m-f2 

3     ''    3.5    +     •    •    •     3.5.7   ....  (3-[-2m— 2)' 

^==-A/o      2.4.6   .    .    .    .(2m+2)v_ 
— 1V'^~3.5.7   .    .    .    .(2m+l)/ 
2.4.6  ....  (2w-f-2)_ 
3.5.6  .    .    !    .  (2m  +  l) 

Ex.  2.     Find  sum  of  ??i  terms  of  -  -f  -^  4.  7i;^T^^  +  etc. 

2  '   2.4  '   2.4.6   ' 

1.3.5.7  .    .    .  (2m+l) 

2.4.6   .    .    .  2m 


Art.    6*     Some  special  modes  of  summation: 

METHOD  OF  DE  MOIVRE. 

Assume  a  series  whose  terms  converge  to  0  and  involve  fJie 
powers  of  an  indeterminate  x;  place  the  series  equal  to  S  and 
midtiply  both  members  of  this  equation  by  a  suitable  binomial, 
trinomial,  etc.,  which  involves  the  powers  of  oc  with  constant 
coefficients;  then  assume  x  so  that  the  binomial,  trinomial, 
etc.,  may  be  equal  to  zero  and  transpose  some  of  the  first  ter)n.^; 


PRINCIPLES  OF  ALGEBRA. 


their  sum  will  be  found  equal  to  the  tium  of  iJie  remaining 
terms. 

Ex.  1.     Let  it  be  required  to  find  tbe  sum  of  ^—^y-; — ,■ 

i.  .  ^  It  .it 

-[-  —j-}-  etc.,  to  infiiiity.       Place  ,S'.^1  j^-r  „  -    74    etc., 

and  multiply  both  members  by  ,r — 1.      We  get 

, .  /       -.  X  x*'     a?     x^ 

.S  (.r— l)--.r-^  ~-f-4-j-r   etc. 

-1-2-3-4-5-  ^*^- 
Whence  by  addition  H  {x~\)=  — l-rY^2"^/3~^3~4"^r5 
-}-  etc. 

Now  suppose  that  x  =^1  and  we  oet  —  4.       _[_  —  \^ — 
^^  °      1-2^2.3^3.4^4.5 

-p  etc.  =  1. 

Here  the  first  term,  —  1,  is  transjDosed  and  is  equal  to 

the  sum  of  the  proposed  series  to  infinity. 

Ex.  2.     If  we  had  multiplied  by  the  binomial  x"^  —  1 

we  would  have  found  the  sum  of  —-_}_--- ^——_j_-    -[-etc. 

i.o      JL.'±      0.0      4.0 

and  of  r—- — 2T— i+TT^ —  etc.,  to  infinity. 
1.3     2.4  '  3.5  ^ 

Ex.  3.     Let  the  multiplier  be  the  binomial  2^  —  1. 

>S'  =  """"^  2  ^  4  "^  4  ~^  ^*^'  *^  i^^^i^y 

2.r— 1 


,S  (2^-lH  2a;-f ^V^l'-f^'  -f  etc 

ir       J?^      r*      ^' 


1 — r  —  -  —  -  —  -  -|-  etc. ;  adding  we  obtain 
2        o        4        5 


.S  (2-^'—l)=— 1+^79-1-2:3 -r374-|-4;5-r  etc.      Now  make 

1  -^ 

'Ix  —  1  =  0,  whence  x  =    ,  and  we  ofet 


2  °  '  1.2.2  ' 

.)  o  o2~fo  !<   03+ F-FTTrf  etc.  =^  0,  and  the  sum  of  the  se- 
2.0.2      o .4: .  Ji      4.5.2 


SUMMATION  OF  SERIES.  7 

Art.  7.  It  will  be  observed  that  when  the  multiplier 
is  a  binomial  the  resulting  series,  before  the  value  of  .r  is 
assigned,  will  consist  of  fractions  having  two  factors  in  the 
denominators;  if  the  multiplier  is  a  trinomial  there  will 
l)e  three  factors,  etc. 

5  6 

Ex.4.     The   sum   of    the     series^  2  3  2'^+ 07472'+ 

7  1 

To  prove  this  let  the  multiplier  be  (2.r — 1)  (.r — 1)  =  2j?' 
— 3.r+l:  then 

.S(ar'-3^+l)  =  2^'+^|-+|:+_+  etc. 

„       Zx^    Z3?    Zx'    Za? 

IT      Hi        ^        "7        nr^ 

4-l+o4""Q"i~V~7  k^Vt."]"  6tc.,  which  by  addition  is 

i5         O  4         O  D 


Bx  ^'^^ 


5x'  6x'  Ix' 


S  (^'-3x+l)  =  1-0+1-273+270+0.  5-f-4.X6+ 
etc.     Now  if  we  make  the  factor  2x — 1  =  0,  we  get  x= 

1  5  6  7  _^ 5 

2^      1.2. 3.  2^+2. 3.4.  2»+3. 4:  5.  2^+  "~1.2.2'        4 

-'4     ' 

Had  we  made  the  second  factor  equal  0  or  ,c  =  1    we 

5'   1         6 
should   have   found  the  sum  of  the  series      ^--^-["0  o  ^4" 

L.  n.  o       ^.0.4 


l-etc.  =f-l       ^ 


3.4.5        "22 

1  X  x^  *  W- 

Ex.    5.        Suppose    --I — 4— — r^;r-  +         ,'  »'    +    ^tc.    =     »S 

and  that  we  use  the  multiplier  ax — h,  we  will  get 

(!!i+J!fL_L  JIfLj.  etc 

\       b        bx  b.v^  bar^ 

V      m     77J-L-/-     m-\-2^     m-{-3r 


0  PRINCIPLES  OF  ALGEBRA. 

And  by  addition  we  find  S  (ax—b)  =  -  -^   i  l'''  +  ''l^~''L^^  - 

m         7n(m-[-r) 
{m^2i-)  a — (m-}-r)6  „ 

— {m^-r)  (m+2r)"  "^  +  ^^^-     "^"^^  ^^'^^^  ^''^  —  6  =  0,  but 
retaining  for  the  present  x  in  the  second  member  we  liave 

alter    transposing the    series     — ^—r- ,- —  x  4- 

m  w(m-fr)  ' 

(m+2r)  ft  —  (??i+r)  b  ^     (m-f  3r)  a  —  (7?i-[-2r)6  3  _ 

(7n-^r)  (m-l-2r)       ^  ^         (?7i+2ry(»i+3r)      ^  ^"  ^^^'^ 

— .     Now  if  any  particular  numerical  series   be    proposed 

and  it  can  be  shown  to  coincide  with  the  above  when  suit- 
able values  are  substituted  for  the  various  letters   we  will 
be  able  to  find  its  sum  from  the  formula. 
Suppose  it  were  required  to  sum  the  series 

o  •  q"1"  Q~f:  •  o2"l~r">  •  03 ~r  ^^^'  Sy  inspection  we  find  that 
L  .  o    o      0.00      ,0  .  /    o 

.r  =  -  and  since  ax  —  b  =  0,  a  =  36;  m  =^  1,   r  =  2  and 
o 

from  the  first  numerator  {ni-[-r)  a  —  mb  =  2.  *.  3a  —  b  =  2; 

18  3  1 

or  3a  —  o  ^  =   2.-.-  «  =  2,  and  a  =    and  />  ^=-.      These 

So  4  4 

several  values  being  substituted  in  the  formula,  term  after 

term,    build    up    the    j^roposed    series,  the  sum  of  which 

6      1      1 
theiefore  =  — =  —  r. 
m     1      4 

The  foregoing  gives  the  sum  of  the  series  to  infinity,  but 
if  it  be  required  to  find  the  sum  of  a  finite  number,  ?i,  of 
terms  we  may  proceed  as  follows : 

-Take  the  series  to  n-\-l  terms,  and  multiply  by  ax — b  as 
l)efore, 

1  X  x^  x"-^  ^       x" 

m.    '    m-\-7'    '    77i-f-2r         '  "  *         in-^{n — !)?■   '^  in-^-nr      * 

ax — b 

ax  ax^  ax""^  ax" 

m        m-{-r~^ ^  m-f{7i — l)r^  m-^rw 

b  bx  bx^  bx"  ,         ,,„, 

{ax — b)S' 


m      m^r      m-\-2r  '    '   '  m-\-nr 


SUMMATION  OF  SERIES.  M 

whence  by  adding  and  transposing  we  get: 
(m+r)a--mb  i,n+^r)a--(rn+r)b^,  +etc.,tonthterm  = 

7n{m^r)        ^        (m-fr)(m+2r) 

b        ax"'^'^ 

SUax — b)  A ■ .     Then,  using  the  values  of  the 

^  '       m      m-^-nr 

letters  belonging  to  the  numerical  series  last  considered, 

13  1 

to  wit:  m  =  l,  r=2,x==^,   ^^^I»   ^^^i'   ^^^  we  have 

f     4.  f2131         41' 

S  =  sum  of  n  terms  of  ^.^  +  —  .^,  +  5^.-  _|_    .    .    .    = 

1  1 


4      4.3«(l+2?i)' 

Ex.  6.     Let  it  be  required  to  find  the  sum  of  the  infinite 

^^"■'^^  1.2.3-4  +  2.3.4*8  +  3X516  +  4.5.6*32  +  ^^''• 

Here  J,  the  square  and  higher  powers  of  which  are  pres- 

ent,  is  represented  by  x  in  the  series  1  -{---|--_j---j-etc. 

2       ij       4 

Moreover,  there  are  thi^ee  factors  in  the  denominators  of 

the  coefficients.     Let  us,  then,  multiply  by  the  trinomial 

ax^ — bx-{-c,  and  we  get 


S{ax'—bx^c) 


,  ,  l^tA.-  l^U/  IjUj  U.//  , 

bx^      bx^      bx^ 
-bx--^--^-^-eic. 

^      ajc^      ax\     , 

«'^  H-^^  +  X+^^'^- 


Whence  adding:     c  -}-  -j-o~^  H ^  n  3      ^'  ^~ 

12a-86-6c        20^156+12c        ^^^    ^^.^^ 

2.3.4  ^         3.4.5  ^        '  y         ' 

S(ax^—bx-^G). 

Now  making  cLic^—bx^G  =  0,  and  in  substituting  that 

value  of  x  =  iy  previously  quoted,  we  get ^  —  2  + c  =  0; 
also,  from  the  first  and  second  numerators  of  the  proposed 


10  PRINCIPLES  OF  ALGEBRA. 

series,  compared  with  those  over  the  same  denominators 
in  the  formulas,  we  get  two  more  equations:  6a — 36+ 2c 
=  19  and  12a— 86+6c'  =  28.  From  these  we  find  a  =  6, 
6  =:  7,  c  =  2,  and  these  placed  in  the  formula,  build  up 
the  proposed  series.   After  transposing  the  first  two  terms, 

19    1         28    1         29     1 
2  and  -3,  we  find:  ^-^-3.^  +  ^J^-^  +  SXsTG  +  '^'^  ^^ 

infinity  =  3— 2  =  1. 

We  may  determine  the  sum  of  n  terms  of  this  series  in 
the  manner  of  the  last  example.     It  is: 

1  4+n 

(7iH-l)(n-f2)2-'- 
To  determine  the  multiplier  to  be  used  for  any  particu- 

lar  case,  assume  the  series  up  to  the  term  — — : . 

n-j-2 

]Sx.  7.     Find  the  sum  of  x-^2x'-{-3x^-^4:x'-\-  etc.,  to  00  . 
Multiply  the  proposed  series  by  x^ — 2a?-}- 1. 

X 

S  = 


1— 2.r-[-^=' 

Ex.  8.     Find  the  sum  of  .r+ 4^^=^-1-9^-1- 16^*+ etc.,  to  00  . 
Use   1 — Sx-{-Sx^ — x^  as  a  multiplier  of  the   series   pro- 


posed. Ans. 


x{l-\-)x 


CHAPTEK    II. 

Elimination. 

Art.  8.  In  order  that  an  equation  may  be  solved,  it 
must  contain  only  one  unknown  quantity.  If  it  is  identical 
it  may  have  an  infinite  number  of  solutions,  but  if  it  is  a 
common  equation  it  will  have  only  a  limited  number  of 
roots.  If  then  we  have  a  single  equation  containing  more 
than  one  unknown  quantity,  and  the  equation  is  not  iden- 
tical, but  is  indeterminate, we  must  attribute  values  to  all  the 
unknown  quantities  save  one,  before  solution.  If  we  have 
two  or  more  simultaneous  equations  we  must  eliminate  so 
as  to  obtain  a  single  equation  with  one  unknown  quantity. 


ELIMINATION  11 

The  methods  which  are  used  when  the  equations  are  of 
the  first  degree  will,  upon  application  to  those  of  the  high- 
er degrees,  be  found  to  fail  to  give  results  practically  use- 
ful. The  method  of  the  greatest  common  divisor  is  that 
usually  employed  with  higher  equations,  and  to  its  discus- 
sion we  will  proceed  after  establishing  a  principle  in  the 
nature  of  equations  which  is 

Art.  9.  If  (I  is  a  root  of  an  equation  luhose  second  viem- 
ber  is  zero,  the  first  member  ivill  be  exactly  divisible  by  the  bi- 
nomial X  —  a. 

Let  the  equation  be  oc'"  -|-  Px""'-^  Qx'""'-^  Rx'""^^ 

4-  Tx  -{-  U=  0,  and  let  a  be  a  root  of  the  equation;  then 
X  =  a  or,  X —  a  =^  0. 

Suppose  the  first  member  to  be  divided  by  ;r  —  a  and 
that  we  continue  getting  terms  of  the  quotient  until  the 
remainder  is  without  an  x,  or  is  independent  of  x.  Let  this 
remainder  z=  II,  and  the  quotient  (which  may  consist  of 
one  or  of  several  terms)  =  Q,  then  {x  —  a)  ^  -f-  i?  =  it"«-{- 

Px"'--'^Qx"'-'-{- -\-Tx-\-V  =^.     But  since  the  second 

member  =  0  the  first  must  =  0,  and  since  a  is  by  supposi- 
tion a  root;  a;  —  a  =  0,  and  this  leaves  i^  =  0.  The  re- 
mainder being  =  0,  the  division  was  exact,  which  proves 
the  theorem. 

The  converse  is  also  true.  If  the  first  member  of  an  equa- 
tion whose  second  is  zero,  is  divisible  exactly  by  the  binomial 
x  —  a,  then  a  is  a  root  of  the  equation. 

Since  the  division  is  exact  there  will  be  no  remainder, 

and  (X  —  a)  Q  =  x'"-{-Px'"-'-\-Qx'"-' j^  Tx  ^  U.     Now 

Avhatever  value  of  x  makes  the  second  member  of  this  equa- 
tion =  0  is  a  root  of  the  proposed  equation;  but  x  ^=  a 
effects  this,  by  reducing  the  first  member  to  zero,  and 
hence  a  is  a  root  of  the  proposed  equation. 

These  properties  belong  of  right  to  the  general  Theory 
of  Equations,  to  the  discussion  of  which  we  will  soon  pass, 
but  being  necessary  to  the  understanding  of  the  principles 
of  elimination  now  to  be  examined  have  been  introduced 
here. 


12 


PRINCIPLES  OF  ALGEBRA. 


Art.  10.  When  two  or  more  equations  are  simultaneous 
they  have  values  for  the  unknown  quantities  entering  them 
which  are  the  same  in  all  the  equations.  These  are  of 
course  common  to  them  all;  and  there  may  be  values  in 
the  different  equations  which  are  not  common.  The  com- 
mon roots  make  the  equations  compatible  and  are  known  as 
compatible  roots. 

If  thus  b  were  a  root  common  to  two  equations  in  x  and 
y,  and  if  b  were  substituted  for  y  in  the  first  members  of 
the  tw^o  equations,  they  would  become  polynomials  in  x 
only;  moreover,  since  b  is  compatible  with  some  value  of  a: 
in  both  the  first  members,  if  we  call  that  value  of  ^  =  a 
then  those  first  members  will  have,  from  Art.  9,  a  com- 
mon divisor,  x  —  a.  Having  substituted  in  the  two  first 
members  the  known  value  of  y,  let  it  be  supposed  that  the 
process  for  obtaining  the  H.  C.  D.  were  applied  to  them; 
it  would  terminate  of  course  in  a  remainder  ^=  0. 

If,  then,  without  substituting  the  value  of  y,  and  even 
without  knowing  it,  we  apply  the  process  for  finding  the 
highest  common  divisor,  we  obtain,  after  a  sufficient  num- 
ber of  operations,  a  remainder  in  y  only.  Now,  if  we  Jmd 
known  and  substituted  the  proper  value  of  y,  this  remainder, 
which  we  will  call  R,  should  be  =  0.  Hence  R=y\i/)  =:  0 
is  a  true  equation.  This  equation  is  called  the  Jinal  equa- 
tion in  y. 

Now,  among  the  roots  of  this  final  equation  in  y  will  be 
found  all  the  compatible  roots  or  values  of  y,  and  when 
they  are  substituted  in  the  last  preceding  divisor  placed 
equal  to  zero,  they  will  give  the  corresponding  values  of  x. 
That  is,  such  will  ordinarily  be  the  result.  But  if  the  pre- 
ceding divisor,  upon  the  substitution  of  the  values  of  y, 
becomes  zero  at  once,  so  that  we  cannot  obtain  the  corres- 
ponding values  of  x,  we  proceed  to  the  next  preceding  one, 
which  wall  be  ordinarily  of  the  second  degree  with  regard 
to  X,  and  give  two  values  of  x  to  each  one  substituted  for 
y.  If  this  fails,  we  proceed  to  the  divisor  last  before  this, 
and  so  on. 

But  usually  the  substitution  of  the  values  of  y,  found  from 


ELIMINATION.  13 

the  final  equation  in  ?/,  in  the  preceding  divisor,  will  not  at 
once  reduce  it  to  zero,  but  will  give  a  polynomial  in  x 
which  will  be  a  common  divisor  of  the  first  members  of  the 
original  equation  after  the  value  of  y  has  been  substituted 
in  them. 

Now,  this  polynomial  in  x  should  be  -equal  to  zero, 
because,  being  a  common  di\isor  of  the  first  mem- 
bers of  the  original  equations,  it  contains  that  factor  of  the 
form  X — a,  or  the  product  of  such  factors,  belonging  to  the 
compatible  values  of  x,  and  probably  other  factors  beside. 
This  divisor,  a  f{x),  might,  if  Ave  knew  the  factors  compos- 
ing it,  be  put  under  the  form  of  {x — ci)f^{x),  or  (,r — a) 
(x — c){x — d)f'(x),  according  to  its  nature,  and  of  course  the 
substitution  for  x  of  the  values,  a,  c,  d,  etc.,  would  reduce 
it  to  zero. 

This  divisor,  then, /(.r)=^  0,  is  a  true  equation,  from 
which  we  ought  to  obtain  the  compatible  values  of  x. 

Art.  11.  Having  obtained  the  final  equation  in  y,  if 
by  factoring  its  first  member,  or  in  any  other  way,  we  can 
solve  it,  we  do  so;  substitute  the  roots  for  y  in  the  last 
preceding  divisor,  or  in  the  one  before  that,  as  the  case 
may  require,  obtain  the  values  of  x,  and  verify  both  by 
substituting  them  in  the  original  equation. 

Art.  12.  Foreign  Roots. — We  will  thus  obtain  all  the 
compatible  roots,  but  we  may  get  also  others.  For  if  in 
preparing  the  dividends  at  any  time,  we  have  found  it 
necessary  to  multiply  any  by  y,  or  any  f{y),  we  may  thus 
have  introduced  foreign  values  of  y,  which  will  appear 
among  the  roots  of  the  final  equation  in  y.  This  might 
have  been  done,  for  instance,  to  avoid  having  y  appear  in 
any  denominator  of  the  quotient.  For  if  we  denote  the 
first  member  of  the  first  equation  by  A  and  of  the  second 
equation  by  B,  and  by  Q  the  quotient  of  A-i-B  and  by  E 
the  remainder,  we  shall  have:  ^1  =:  0,  i?  =  0,  and  A  == 
BQ^R,  and  from  the  next  division:  B  =  RQ'^R',  R  = 
-K'  Q  "+  ^",  6tc.  Now,  since  the  equations  are  simul- 
taneous and  their  first   members   have   a   common   divi- 


14  PRINCIPLES  OF  ALGEBRA. 

sor,  the  remainders  R''\  E'' ,  B' ,  etc.,  will  on,  substituting 
the  values  of  y  and  x  be  found  successively=  0,  and  finally 
R  ^=  0.  Now,  if  Q,  or  any  quotient,  should  have  y  in  its  de- 
nominator the  substitution  of  its  value  in  such  denomina- 
tor might  reduce  it  to  0  and  make  (^>  =  oo  ,  and  then  al- 
though B  =  0,  BQ  would  not  be  ^^  0.  The  supposition 
on  which  the  process  is  founded  is  that  A  and  B  are  al- 
ready (or  have  been  made)  whole  with  respect  to  y. 

In  this  way  foreign  roots  may  have  been  introduced  into 
the  final  equation.  They  may  be  detected  by  trial  in  the 
proposed  equations  and  rejected. 

Furthermore,  it  may  happen  that  all  the  proper  values 
may  not  be  found  by  means  of  the  final  equation,  since  we 
may  have  suppressed  some  factors  in  the  process  for  find- 
ing the  H.  C.  D.,  which  w^ould  reduce  to  zero  on  the  substitu- 
tion of  the  proper  value  of  y .  Such  factors  should  be 
placed  :=  0  and  the  values  of  y  substituted,  and  the  values 
of  X  found  from  the  resulting  equations. 

If  the  final  remainder  should  be  independent  of  y,  it  of 
course  is  not  zero,  and  the  equations  have  no  compatible 
roots. 

Art.  13.  Mention  has  already  been  made  of  a  solution 
of  the  final  equation  in  y  when  it  was  possible  to  detect  its 
factors.  Similarly  when  w^e  can  detect  the  factors  of  the 
first  members  of  the  proposed  equations  we  may  shorten 
the  process.  Suppose  all  the  factors  of  each  to  have  been 
discovered:  they  will  be  of  tw^o  kinds,  commoyi  and  those 
not  common.  Among  the  common  some  may  be  altogether 
in  X,  some  altogether  in  y  and  some  functions  of  both  x 
and  y. 

Likewise  the  same  three  species  may  exist  among  the 
factors  not  common. 

Now  any  one  of  these  factors  being  placed  =  0  will  sat- 
isfy the  equation  to  which  it  belongs.  Suppose  we  first 
consider  those  which  are  common  to  the  two  first  members. 
Those  in  x  only  will  give  a  limited  number  of  values  for  x 
and  any  values  whatever  for  y  provided   they   are   finite; 


ELIMINATION.  15 

those  in  y  only  will  give  a  limited  member  of  values  for  y 
and  leave  .r  indeterminate,  and  those  which  are  functions 
of  X  and  y  both  will  give  an  infinite  number  of  sets  of  val- 
ues for  .r  and  y. 

Second.  Of  those  not  common  we  cannot  place  two  in 
X  only,  or  in  y  only,  equal  to  zero  at  the  same  time,  for  it 
would  not  be  true  unless  one  was  equal  to  the  other  multi- 
plied by  some  constant  factor.  And  this  is  contrary  to  the 
supposition  that  we  have  already  considered  all  that  were 
common.  There  remain  those  not  common  and  which  con- 
tain both  X  and  y.  To  these  we  should  apply  the  process 
for  the  H.  CD.,  and  we  will  obtain  a  limited  number  of 
values  for  x  and  also  for  //. 

EXAIMPLES. 

1 .  Let  the  equations  be  x^  — ^yx'^-}-{3y^ — Sy-\-l)x — 2/*+ 
y'^ — 2y=^0,  and  x^ — 2«/.r~j-]/'* — ^=0. 

First  Division. 

of^—Syx'-i-(Sy'-y-i-l)x—f-{-y'—2y  \  x'—2yx+f—y 
x^ — 2yx'^-{-{y^ — y)x  \      x — y 


—yx'^{2y'^l)x—f-{-y'—2y 
—yx^-{-2y^x         —y'+y'' 

This  division  was  performed  without  preparation.     So, 
likewise,  with  the 

Second  Division . 

x^ — 2yx-]~y'^ — y  \  x — 2y 
x"^ — 2yx  •         I  ^ 

y'^ — y     and  y"^ — y=0  is  the  final  equation  in  y . 
Its  roots  are  .v=0  and  y=l .     Hence  we  have  the  systems 


and 


?/=l 
x=--2 


2.    Let  the  equations  be  oif-{-y^=0,  and  x^-{-xy-\-y^ — 1=0 
No  preparation  will  here  be  necessary. 


16 


PRINCIPLES  OF  ALGEBRA. 


First  Division. 

x^-\-x'^y-\-y'^x — X        |        x — y 
~yx^—y\x^x-{-y^         ~~  " 

—yx^—y^'x—y^-i^y 
x-^2y'—y 

Second  Division. 

x^J^yx-{-yl—l  I  x^2y'—y 

x^—yx^2y^x  \    x-\-2y — 2y^ 

(2y—2y')x-\-y'—l  ''        ^ 

{2ij-2f)x-lf^6,/-2f 

4:i/^ — 6y+3/ — 1  and  this  placed  =  0  gives  the 
final  equation  in  y. 

It  is  evident  upon  inspection  that  y  =  l  and  y  =  — 1  are 
roots  of  this  equation,  and  the  other  four  are  ±:J;/ldz|/  JZ3] 

y  =  1  and  y  =  — 1  give,  upon  substitution  in  the  origi- 
nal equations,  ,t  =  1  and  ^=—1,  values  to  be  expected,  as 
the  equations  are  symmetrical. 

3.  Let  the  equations  be:  x^-\-2yx^-\-2y{y — 2)x-{-y*  —  4 
=  0sLndx^^2xy-{-2y'—5y-^2  =  0. 

First  Division. 

x^-j-2x^y^2y^x—4:yx^y^—4:   \  x''-^2xy^2y'—^y^2 

a^-\-2x^y-]-2y'^x — 5yx-{-2x        \  x 
(2/— 2)^+17^— 4 
and  this  remainder  may  be  factored  thus,  {y — 2)[.r-j-2/-l-2], 
and  the  factor  y — 2  laid  aside. 

Second  Division. 

x'-^2yx^2^y^—^y^2  \  x-^y^2 


xf^r  !/-^+2^ I  x^{y—2) 

(y—2)x^2y^—by^2 
{y—2)x-^  y^        —4 
2/^— 52/+6 
which,  placed  =  0,   gives   y  =  2  and  i/  =  3.     y  =  2  gives 
x  =  Q  and  x  =  — 4.       y  =  ^   gives   xt= — 1    and   x  =  — 5 


TrENERAL  THEORY.  17 

from  the  second  equation,  but  upon  trial  with  the  first 
equation  only  the  roots  x  =  — 4,  y  =  2,  or  ^=  — 5,  y  =  d,  are 
found  to  be  compatible  roots. 

'  The  suppressed  factor  y  —  2  gives  ?/  =  2;  a  value  also 
found  from  the  final  equation. 

The  foregoing  treatment  of  this  subject  is  mainly  taken 
from  the  excellent  discussion  of  Elimination  in  Hackley's 
Algebra. 

Art.  14.  Labatie  and  Sarrus  have  perfected  a  method 
of  elimination  b}^  which  foreign  roota  are  not  introduced  into 
the  final  equation.  This  mode  is  quoted  by  Hacldey  and 
by  Todhunter  in  his  Theory  of  Equations;  but  it  is  doubt- 
ful whether  any  advantage  is  gained  over  the  simplicity 
and  ease  of  trying  the  roots  in  the  original  equations  and 
rejecting  such  as  do  not  verify  them. 


CHAPTER    V. 

Nature  or  General  Theory  of  Equations. 

Art.  15.  An  equation,  as  we  know,  is  an  algebraic  ex- 
pression of  the  equality  of  two  quantities.  (And  before  it 
can  be  solved  must  contain  a  single  unknown  quantity.) 
This  statement  is  true  even  of  an  identical  equation  which 
is  true  for  any  value  of  the  unknown  quantity  or  quantities 
entering  it;  for  the  mind,  in  the  act  of  attributing  a  value 
to  any  unknown  quantity  in  the  equation,  may  be  sup- 
posed to  regard  that,  for  the  time  being,  as  the  only  one . 

Now  the  two  equal  quantities,  i.  e.,  the  two  members,  of 
every  equation,  may  be  placed  in  the  first  member  leaving 
the  second  member  zero;  and  the  polynomial,  [after  it  has 
been  arranged  with  reference  to  the  descending  powers  of 
the  unknown  quantity,  may  be  divided  by  the  co-efficient 
of  the  highest  power,  and  so  at  the  same  time  may  be  the 
second  member,  placing  the  equation  in  the  form 
3 


18  PHINCIPLES  OF  ALGEBRA. 

j.-_j_P;^— ^_I_g.r— _i^i^./;"'--|-  .  .  .  ,-^Tx^  U=  0.  .  .  .(1) 

This  general  form,  which  is  often  called  the  reduced  form, 
has  the  co-efficient  of  x'"  unity  and  P,  Q,  B,  T,  etc.,  any 
quantities  not  transcendental;  they  maybe  algebraic  or  nu- 
merical, whole  or  fractional,  positive  or  negative,  rational, 
irrational,  real,  imaginary  or  zero.  When  any  co-efficient 
is  zero  the  corresponding  power  of  the  unknown  quantity 
is  usually  absent.  The  equation  is  ihen  incomplete;  but 
when  all  the  powers  are  present  from  the  highest  to  the 
zero  power  the  equation  is  complete. 

The  co-efficient  of  the  zero  power  of  the  unknown  quan- 
tity is  called  the  absolute  tei^m  of  the  equation. 

Art.    16.     The  form/  (x)  =  x"'^Fx"'-'^ Qx'"-'^ + 

Tx-\'U'=  0  above  described  is  the  most  convenient  for  ex- 
amining the  nature  of  equations,  but  many  of  the  proper- 
ties of  equations  which  will  be  demonstrated  are  true  when 
the  equation  has  not  been  reduced  to  this  form. 

And,  on  the  other  hand,  many  proijerties  will  be  demon- 
strated only  of  equations  having  real  co-efficients  and  even 
of  those  having  their  co-efficients  numbers. 

Art.  17.  Every  equation  has  at  least  one  root.  Much 
ingenuity  and  mathematical  skill  have  been  used  in  de- 
monstrating this  proposition  by  algebraic  analysis,  but  it 
seems  unnecessary  for  it  is  almost  if  not  quite  axiomatic. 
Since  an  equatio;i  is  an  algebraic  expression  of  the  equal- 
ity of  two  quantities,  or  of  the  fact  that  their  difference  is 
=  0,  there  mast  be  some  qaantity,  or  value  of  the  unknown, 
such  that  when  its  different  powers  have  been  multiplied 
by  the  appropriate  co-efficients  and  the  sum  of  all  the  pro- 
ducts taken,  the  result  shall  be  zero.  OtJiei-wise  there  would 
be  no  equation;  the  truth  would  not  have  been  told  by  the 
algebraic  expression. 

The  requisite  value  of  the  unknown  quantity  may  be  a 
real  quantity  or  an  imaginary  expression ;  and  it  is  called  a 
root  of  the  equation. 


GENERAL    THEORY.  19 

Art.  18.  Every  equation  of  the  mth  degree  has  m  roots 
and  no  more. 

We  have  just  seen  that  every  equation  has  at  least  one 
root,  and  we  already  know  that  an  equation  of  the  first  de- 
gree has  one  root;  also,  that  an  equation  of  the  2d  degree 
has  two  roots,  and  it  is  now  to  be  proved  that  an  equation 
of  the  mth  degree  has  m  roots;  that  is,  the  number  of  roots 
is  equal  to  the  number  of  units  in  the  exponent  which 
shows  the  degree  of  the  equation.  It  has  been  proved  in 
Art.  9  that  if  a  is  a  root  of  an  equation  the  second  mem- 
ber of  which  is  zero  the  first  member  will  be  exactly  divis- 
ible by  .r  —  a. 

Now  suppose  that  a  is  a  root  of  the  equation 

af'-J^Px^-'^Bx^'-'-^-Bx^-^'-] .     -fTar+r=0,  then   we 

shall  have  x'"-{-Px'"-'-i-  Qx""-'-]- Tx^  U^{x  —  a)  [x'^-'-i- 

PV-^4- ...._!_  T'x^  U']....  (1) 

Now  this  can  be  satisfied  by  placing  x  —  a  =  0;  and  also 

by  placing  a^^'-^-fP'^p"'-"-] -^T'x-\-U'  =  0,  which  is 

a  new  equation,  and  it  also  has  at  least  one  root. 

Suppose  that  this  root  is  6,  then,  as  before,  we  have  x*^~^ 
_^p.^^-i_l_  _  _  _  ^j^T'x  f  U'=  {x—b)  \x^'-'-^P''x^'-^^Q''x^-' 
-j-  •  . .  -f  T'^x-j-  f/"]  which  can  be  satisfied  by  placing  x — b 
=  0;  and  also,  by  placing  ^— -fP^'j:— ^-f- . . .  .  -f-  T''x^  W 
=  0;  and  this  is  a  new  equation  having  at  least  one  root, 
which  may  be  called  c,  and  when  the  corresponding  factor 
X — c  is  divided  out  we  shall,  as  in  the  previous  cases,  have  a 
new  equation.  The  degree  of  this  equation  will  be  m  —  3. 
Continuing  this  process  until  the  original  first  member  has 
undergone  m  —  1  successive  divisions  we  shall  have  a  quo- 
tient of  the  first  degree,  of  the  form  x  —  Z,  which,  placed 
equal  to  zero,  gives  an  equation  of  the  first  degree,  with  one 
and  but  one  root.  Thus  the  total  number  of  roots  is  r», 
and  the  continued  product  of  the  corresponding  factors 
formed  by  subtracting  each  root  from  x  will  be  equal  to  the 
original  first  member,  so  that  we  shall  have  the  equation 


20  PRINCIPLES  or  ALGEBRA. 

X — c)  {X — d) (;r — /)    ...  (2) 

And  there  can  be  no  more  roots  than  m;  for  if  there 
could  be  another  and  it  were  h,  different  from  a,  6,  c,  d. . 
ly  there  would  be  a  factor  x — k  which,  multiplied  into  the 
product  of  all  the  others,  would  give  for  the  first  mem- 
ber a  different  polynomial,  and  one  of  a  degree  higher 
by  unity;  hence  k  vannol  he  a  root. 

This  fact  may  also  be  seen  thus:  If  k  is  a  value  of  .r, 
let  it  be  substituted  in  the  continued  product,  {x — a){x — h) 

ix — 0) (X — I)  =  0,  and  we  derive  ik — a){k — b) 

{k — c) ik — /),   which   cannot  be  zero,  because 

none  of  the  factors  are  zero;  whereas,  when  a  true  root,  as 
a,  b,  c,  etc.,  is  substituted,  there  will  always  be  one  factor 
which  vanishes.     Thus  the  theorem  is  seen  to  be  true. 

Art.  19.  Equal  Boots. — It  may  happen  that  one  or 
more  of  the  factors  x — a,  x — h,  etc.,  shall  be  repeated,  in 
which  case  the  corresponding  roots  will  appear  as  often  in 
the  equation;  these  are  called  equal  roots.  Thus,  (r — a)^ 
(dp — b)ix — c)'^  =  0  is  an  equation  of  the  6th  degree,  which 
has  three  equal  roots,  a,  and  two  equal  roots,  c. 

It  will  be  shown  further  on  that  when  an  equation  has 
equal  roots  they  may  be  discovered  and  the  first  member 
divided  by  the  product  of  the  factors  belonging  to  them, 
thus  depressing  or  reducing  the  degree  of  the  equation. 
This  operation  is  spoken  of  as  "  dividing  out"  the  roots. 

COMPOSITION    OF    EQUATIONS. 

Art.  20.  When  the  roots  of  an  equation  are  a,  b,  c,  d, 
6,  ....  Z,  we  have  seen  that 

af"-\-Pii(^''~'^Qx/''-''-^Tx-\-U  =  {x—a)^x—b){x—G)   .  .  (x—l) 

..'....    (1) 
Now,  if  the  multiplications  indicated  in  the  second  member 
be  performed,  the  result  will  be  as  follows: 


GENERAL   THEORY. 


21 


xr-\-x' 


\—h 


ab  -\-  etc. ,  -f  X'' 

-{-ac 

-\-ad 


-f6c 
-\-hd 


bS 


ztabcd. .  +  •  • + (ibcde. . 
±:abce  . . 
±abcf .  .o        ri" 


ztbcde 


ft)         ,__ 


bcdf..^        S- 


CO 


and  since  the  equation  of  which  this  is  the  second  member 

is  identical,  we  have,  from  the  principle  of  Indeterminates' 

Coefficients : 

P  =  —a—b—c—d  .  .  —k—l;  or,— P  =  a+Z>+c  .   .  +A:-f-/. 

g=a6+ac-f  .   .  -i-bc^  .   .  -\-kL 

E=: — abc — abd—  .   . — ikl;  or, — R  =  abc-{-abd-\-  .   .  -]-ikl. 

S==abnd-\~abce-\-  .    .    .  gikl. 


U=  ±iabcde  ....  kl;    or,  =fi  U=  abode  .  .  .  .  kl. 

The  quantities  a,  b,  c,  d,  e  .  .  .  .  k,  I,  all  appearing  with 
the  negative  sign,  the  product  of  an  even  number  of  them 
is  plus  and  of  an  odd  number  minus,  which  accounts  for 
the  double  sign  wherever  it  appears,  because  in  those  cases 
the  number  of  factors  is  not  known. 

From  these  results  the  following  important  relations  of 
the  roots  of  an  equation  to  its  coefficients  are  manifest,  to 
wit; 


22  PKINCIPLES  OF  ALGEBRA. 

First.  The  co-efficient  of  the  second  term  (with  its  sign 
changed)  is  the  algebraic  sum  of  the  roots. 

Second.  The  co-efficient  of  the  third  term  is  the  sum 
of  the  combinations  of  the  roots  in  groups  of  two . 

Third.  The  co-efficient  of  the  fourth  term  (with  its 
sign  changed),  is  the  sum  of  the  combinations  of  the  roots 
in  groups  of  three;  and  so  on. 

Fourth.  The  co-efficient  of  the  absolute  term  (with  its 
sign  changed  when  it  is  even  numbered,  i.e.,  when  the  de- 
gree of  the  equation  is  odd),  is  the  continued  product  of 
the  roots . 

Art.     21.  DEDUCTIONS . 

Since  the  absolute  term  is  the  product  of  the  roots  it  will 
be  exactly  divisible  by  any  root;  and,  also,  when  there  is 
•no  absolute  term  one  of  the  roots  is  zero .  Further,  when 
there  is  no  second  term  it  is  because  the  sum  of  the  posi- 
tive roots  is  exactly  equal  to  the  sum  of  the  negative  roots. 

Art.  22.  When  the  roots  of  an  equation  are  all  posi- 
tive, the  terms  will  be  alternately  positive  and  negative; 
because  the  product  of  an  even  number  of  negative  terms 
is  plus  and  of  an  odd  number  is  minus . 

Art.  23.  Since  the  first  member  of  an  equation  of 
which  the  second  =  0  is  composed  by  multiplying  together 
the  factors  {x — a),  {.r — b),  etc.,  it  will  have  m  factors  or  divis- 
ors of  the  1st  degree;  and  since  any  two  of  them  may  be  mul- 
tiplied together,  giving  a  factor  of  the  2d  degree, any  three 
giving  a  factor  of  the  third  degree,  and  so  on,  there  will  be 

m(m — 1)^.  .            .  ,,     ^T  ,             m(m—l){ni — 2\        . 
-5^-^— Mivisors  of  the  3d  degree,— ^^ —     '  \ ^  divisors 

of  the  4th  degree,  and  so  on. 

Art.  24.  When  a,  b,  c,  etc.,  are  the  roots  of  an 
equation  that  equation  is  (x — a){x — b){x — c).  .  .  .(x — 1)=0. 
Suppose  the  roots  of  an  equation  are  1,  2,  3,  4:  the  equa- 
tion is  (x—l){x—2)(x—3)(x—^)=x'—10x''^S5x'— 50x^24: 
=0. 


GENERAL  THEORY.  23 

EXAMPLES. 

1.  Form  the  eciiuitiou  whose  roots  are  3,  7  and  —  6. 

2.  Form  the  equation  in  which  the  roots  are  9,  5,  —  1, 
and  —  3 . 

3.  What  is  the  equation  of  which  the  roots  are  —  3, 
2+  1    —1  and  2—1—1  ?     Ans.  a^—x'—l.r~\- 15=^0. 

Art.  25.     Since  in  the  reduced  equation    U=abcde .  .  ./, 
T=abcde.  .k-^-ahcde.  .h-^-ahcde.  .g,  -f  etc. ,  where  the  terms 
of  the  value  of  7" are  composed  of  m — 1  letters  each,  if  we 
divide  the  latter  by  the  former  we  get 
T     abcde  ....h     ,  abode.... g    ,      ^  1111 

u=abrde:::.m^^d;^d^:z:gi^'^'-=^b^-<i^d:^  '^'- 

Therefore,  the  co-efficient  of  the  last  but  one  divided  by 
the  absolute  term  is  the  fUdii  of  the  reciprocals  of  the  roots. 

S'       1        1 

In  the  same  wav  it  may  be  shown  that  -=^=— r-  -U  —  -4- 

U      ah  ^  ac   ^ 

-—   I    , — 1-  etc.,  wherein  S  is  the  coefficient  of  x\ 
ad  '   be 

Art.    26.     If  the  coefficients  of  an  equation  ar-e  whole  num- 
bers, no  root  can  be  an  exact  fraction. 

For,  suppose    -,    an  irreducible  fraction,  to  be  a  root  of 


.r'"  iPx'""'^Qx'" 

-'+ 

.    .    .  +7k-+r=0; 

then,  since- =.r, 
b 

a'"           a'"-'    , 

+  Tl+U=0;    . 

a'" 

0 

Qa""-'h~lta"'-'l)''- 

—     .       . 

.   .   .  —Tab'"-'— lib'" 

~'.       The  second 

a"' 
member  is  whole,  and       is  a  fraction,  since  b  is  supposed 

to  be  prime  to  a,  and  therefore  to  a"\  Here,  then,  is  an 
absurdity  of  a  fraction  equal  to  a  whole  number,  which 
establishes  the  proposition . 

Art.  27.  The  imaginary  roots  of  an  equation  enter  by 
pairs,  when  the  coefficients  are  real;  and  if  the  coefficients  are 
rational,  all  roots  which  are  not  rational  enter  by  pairs. 


24  PRINCIPLES  OF  ALGEBRA. 

r 

Imaginary,  or  impossible,  roots,  as  they  are  sometimes 
called,  are  cases  of  the  general  form,  a-\-y' — 6'^  and 
a — \/—lf.  These  give  the  factors  {x — a~\/ — lf)(x — n-j- 
|/ — })^)r=.r'^ — 2aa^-[-a'^-j^6^  The  product  of  the  imaginary 
roots  is  (a-}- 1/ — ^^)(« — v' — ¥)  =  d^^l)^\  or,  (a-f-i  — h)(a — 
1/ — h=^a^-\-h;  both  real,  rational  and  positive.  The  prod- 
uct of  irrational  but  real  roots:  (a-\-yh){a — ^  'h)=^d^ — />, 
which  is  real  but  not  necessarily  positive.  If  either  of 
these  be  substituted  in  an  equation  for  j",  it  is  apj^arent 
that  the  results  w^ill  be  partly  real  and  parti}  imaginary, 
unless  some  of  the  coefficients  could  furnish  the  necessary 
factor  to  make  the  imaginary  quantities  disappear  from  the 
product.  But  the  coefficients  in  this  article  are  supposed 
to  be  real.  Consequently  there  must  be  another  imaginary 
root  of  the  proper  form,  to  cause  the  product  to  be  real. 
If,  now,  a  thiril  imaginary  root  should  enter  into  the  com- 
position of  the  equation,  a  fourth,  and  of  the  necessary 
form,  must  enter  to  keep  the  product  real.  There  cannot 
he,  therefore,  an  odd  number  of  imaginary  roots. 

If  the  coefficients  are  further  supposed  to  be  all  rational, 
it  is  evident  by  the  same  course  of  reasoning  that  all  irra- 
tional roots  must  enter  by  pairs. 

Art.  28.  Hence  evert/  equation  of  an  odd  degree  has  at 
least  one  real  root,  with  a  sign  different  from  that  of  the  ab- 
solute term.  The  imaginary  roots  are  of  the  form  a  -f 
|/ — t)^  and  a — |/ — 6%  or  else^  '^+v^ — b  and  a  —  y  ' — b,  and 
their  products  result  in  the  sums  of  positive  quantities. 
And  this  positive  sum  is  a  factor  of  the  absolute  term  and 
exercises  no  influence  on  the  sign  of  that  term .  And  so  of 
the  product  of  all  of  the  imaginary  pairs .  This  leaves  the 
one  real  root  to  give  sign  to  the  absolute  term  which,  of 
course,  is  the  opposite  of  its  own.     (See  Art.  20.) 

It  is  also  true  that  every  equation  of  an  odd  degree  hav- 
ing rational  co-efficients  will  have  at  least  one  rational  root; 
the  sign  may  or  may  not  be  the  same  as  that  of  the  abso- 
lute term. 


GENERAL    THEORY.  25 

Art.  29.  Erery  cquaiion  of  c/ven  degree  and  having  real 
ro-effieien(f<,  with  ifs  abi^olufe  term  negative,  will  have  at  leant 
tivo  real  rootn,  one poHitire  and  the  other  negative.  The  pro- 
ducts of  the  pairs  of  imaginary  roots  will  exert  no  inflneu.^e 
on  the  si.L>n  of  the  absolute  term,  and  if  all  the  7'oots  were 
imaginary  the  absolute  term  would  he  pot^itive,  but  as  it  is 
not  positive  there  must  be  at  least  two  real  roots  and  such 
that  their  product  will  be  negative,  they,  therefore,  must 
have  diiferent  signs. 

Art.  30.  Every  equation  will  have  an  even  number  of 
real  positive  roots  if  the  absolute  term  is  positive;  and  an  odd 
number  of  sueh  roots  if  the  absolute  term  is  negative. 

First.  When  the  degree  is  even  and  the  absolute  term 
positive.  The  degree  being  even  the  number  of  the  abso- 
lute term  is  odd,  and,  therefore,  it  is  the  continued  product 
of  the  roots  just  as  it  stands.  In  this  case  that  product  is 
positive.  If  there  are  any  imaginary  roots,  the  quadratic 
factors  belonging  to  the  pairs  will  exert  no  influence  on 
the  sign  of  the  absolute  term .  The  total  number  of  roots 
being  even,  the  number  of  real  roots  must  be  0  or  even . 
Now  the  product  of  the  real  roots  must  be  positive  and  if 
there  is  any  real  root  negative  there  must  be  another  one 
negative  to  neutralize  the  influence  of  the  sign,  otherwise 
the  sign  of  the  absolute  term  would  be  changed.  Hence 
the  number  of  the  real  positive  roots  is  even,  which  proves 
the  theorem  for  this  case. 

It  is  apparent  that  the  number  of  negative  real  roots 
would  also  be  even. 

Second.  When  the  degree  is  even  and  the  absolute 
term  negative.  Here,  also,  the  absolute  term,  as  it  stands, 
is  the  product  of  the  roots;  and  if  there  are  imaginary 
roots  they  exert  no  influence  on  the  sign.  The  number  of 
real  roots  is  even,  and  since  the  product  is  negative  there 
must  be  at  least  one  which  is  negative.  This  may  be  con- 
sidered as  set  aside  for  the  moment.  There  now  remain  for 
consideration  an  odd  number  of  real  roots,  whose  product 
must  be  positive,  and  if   there  is  among  these  a  negative 

4 


26  PRINCIPLES    OF    ALGEBRA. 

root,  there  must  also  be  another  of  the  negative  sign  to 
neutralize  its  effect;  in  other  words,  if  there  aie  any  nega- 
tive roots  among  those  dov^^  being  con.-idered,  there  must 
be  an  even  number  of  them;  consequently,  the  number  of 
real  positive  roots  is  odd. 

Third.  AVhen  the  degree  is  odd  and  the  absolute  term 
positive.  In  this  case  the  sign  of  the  absolute  term  must 
be  changed  to  give  the  continued  product  of  the  roots;  that 
is,  the  product  is  negative.  We  know  (Art.  28),  that  for 
such  an  equation  there  is  one  real  and  negative  root.  Let 
that  be  set  aside  ns  necessary  to  change  the  sign  of  the  ab- 
solute term.  Th^^  total  number  of  roots  remaining  is  even, 
and  the  number  of  real  roots  remaining  is  even .  More- 
over the  product  of  these  remaining  roots  must  be  positive 
and  consequently  if  there  are  among  these  any  negative 
roots  there  must  be  an  even  number  and  therefore  the 
number  of  real  positive  roots  must  be  even . 

Fourth.  When  the  degree  is  odd  and  the  absolute  term 
negative  (By  Art.  28)  there  is  one  real  positive  root.  The 
total  number  remaining  is  even,  and  the  number  of  real 
roots  remaining  is  even;  but  their  product  must  be  posi- 
tive as  the  real  root  set  aside  is  positive;  consequently 
among  these  remaining  real  roots  if  there  are  any  negative 
roots  there  must  be  an  even  number  of  theui,  likewise  an 
even  number  of  positive  real  roots,  which,  with  the  one  set 
aside,  makes  the  number  of  real  j^ositive  roots  odd.  Thus 
the  theorem  is  established. 

Art.  31.  //  the  signs  of  the  alternate  terms  of  an  equa- 
tion be  changed,  the  roots  of  the  new  equation  ivill  be  the  same 
as  those  of  the  former  equation  but  with  opposite  signs. 

Let  the  equation  be  .r"^Px"'-'-^Qx"'"^^  .  .  .  .^Tx-\-U 
=^0...  (1)  If  we  change  the  alternate  signs,  beginning 
with  the  second,  we  have  .r-— P.r'"-^-f  ^.r— "'-^  .  ^-Tr±U 
=0....(2).,  and  beginning  with  the  first,  we  have 
— x»'^Px"'"'^ — Qx"*"'^^ dz  Tx^  U~0 (3),  which  equa- 
tions, (2)  and  (3),  are  merely  one  and  the  same.  Now  sup- 
pose -\-a  to  be  a  root  of  (1)  and  to  be  substituted  in  it  for  x. 


GENERAL  THEORY.  27 

The  result  will  be  a'"^Pa"'-'-\-Qa'"-'-{- .  :..^Ta^U=Q.  (4). 
Now  if  ^  rt  is  a  root  of  (2)  and  (3)  it  must,  on  'substitution 
in  the  one  or  in  tlie  other  (as  may  be  suitable,  for  they  are 
merely  two  forms  of  the  same  thing),  give  eq.  (4). 

When  m  is  even  use  (2)  and  when  ??i  is  odd  use  (3) . 
Therefv^re  —a  is  a  root  of  (2)  or  else  of  (3).  Hence  the 
principle .  Changing-  the  signs  of  all  the  terms  would  not 
affect  the  roots,  since  it  would  simply  be  multiplying  both 
members  of  the  equation  by  — 1. 

•  DESCARTES'     RULE. 

Art.  321.  No  equation  van  have  more  positive  roots  than 
there  are  variations  in  the  signs  of  its  terms,  nor  more  nega- 
tive roots  than  there  are  permanences  of  those  signs . 

To  demonstrate  this  assume  the  equation  x"'±lPx"'~'±: 
Q.r'"-'±E.r"'~^±  .  .,±Tx±:  U=0;  in  which  the  signs  come  in 
any  order  that  may  be  prescribed.  Now  suppose  that  we 
introduce  one  more  positive  root,  which  will  be-  done  by 
multiplying  by  x — a,  and  note  the  effect  on  the  signs. 
The  product  will  be 


±F  i  x^±Q 


— a       :T^Pa 


,w— / 


±E 


,r'«-'zb ±U\  X 

H=r    1  =pC7a=0, 


Now  so  long  as  the  co-etficient  in  the  upper  line  is  greater 
than  the  one  in  the  lower  line  it  will  determine  the  sign 
of  the  total  co-efficient  of  that  term;  if  we  suppose  then  in 
the  first  case,  that  all  the  upper  were  greater  than  those  be- 
low we  would  have  the  same  number  of  variations  and  per- 
manences as  in  the  original  equation,  but  having  to  come 
down  at  last  to  =p  Ua,  there  is  one  more  variation  than  in 
the  original  equation.  If  the  lower  co-efficients  are  all 
greater  than  those  above  they  will  give  sign  to  the  terms;  but 
the  signs  froin  the  second  toward  the  right,  being  always 
the  opposite  of  the  signs  of  the  original  first  member,  the 
number  of  changes  of  sign  and  of  permanence,  or  repeti- 
tion of  sign,  will  be  the  same.  But  one  more  variation 
was  introduced  when  we  descended  at  the  second  term . 

When  a  co-efficient  in  the  lower  line  is  affected  with  a 
sign  coi4rary  to  the  corresponding  one  above  and  is  also 


28  PRINCIPLES  OF  ALGEBRA. 

greater  than  that  above,  there  is  a  change  from  a  perma- 
nence of  sign  to  a  variation,  for  the  lower  co-efficient 
gives  sign  to  the  term,  and  we  know  that  it  is  different 
from  that  of  the  preceding  term  above  which  is  here  sup- 
posed to  be  the  same  as  that  of  the  co-efficient  above  in 
this  term .  Hence  each  time  we  descend  to  the  low^er  line 
in  order  to  determine  the  sign  there  is  a  variation  which  is 
not  found  in  the  original  equation,  and  if,  after  descending, 
we  remain  in  the  lower  line  throughout,  the  number  of 
permanences  and  variations  of  sign  henceforth  will  be  the 
same  as  in  the  given  equation  because  the  signs  are  al- 
ways the  opposite  of  those  above .  If  we  ascend  again  to 
the  upper  line,  we  might  make  either  a  permanence  or  a 
variation;  but  suppose  the  worst,  and  that  always  there 
would  be  a  permanence,  it  would  merely  offset  the  variation 
gained  in  coming  down,  and  it  will  be  necessaiy  to  come 
dow^n  at  last,  making  a  variation  at  that  time. ,  Therefore, 
the  effect  has  been  to  produce  one  more  variation  than  the 
original  equation  had;  and  so  it  would  be  upon  the  intro- 
duction of  every  positive  root. 

Similar  reasoning  would  show  that  the  multiplication  by 
the  factor  x-\-a,  belonging  to  a  negative  root,  would  neces- 
sarily introduce  a  permanence  of  sign .  And  since  the  in- 
troduction of  every  positive  root  brings  a  variation,  and  the 
introduction  of  every  negative  root  brings  a  permanence, 
the  Rule  of   Descartes  is  shown  to  be  true . 

Art.  33.  When  the  roots  are  all  real  the  number  of 
positive  roots  Avill  be  the  number  of  variations,  and  the 
number  of  negative  roots  the  number  of  permanences . 

Suppose  that  the  degree  of  the  equation  was  m;  then, 
the  complete  number  of  terms  being  ?7i-]-l.  and  n  represent- 
ing the  number  of  variations  and  p  the  number  of  perma- 
nences, m=n-\-p. 

Again,  suppose  that  k  =  the  number  of  positive  roots 
and  r  =  the  number  of  negative  roots.  We  shall  have : 
m  =  k-^r;   hence  n-^-j^  =  k-\-r,  and  n — k  =  r—^.     Now, 


GENERAL    THEORY.  29 

by  Descartes'  Kule,  k  cannot  be  >/i;  nor  can  it  be  less, 
because  that  would  make,  in  the  second  member,  r^p. 
which  the  Rule  forbids?     Therefore  n  =  k  and  p  :=::!  r. 

DE    GUa's    criterion. 

Art.  34.  If  a  term  of  an  equation  is  absent  between  two 
lerins  Jiarlng  like  signs,  there  are  two  imaginary  i^oots. 

The  absent  term  having  0  for  a  co-efficient,  we  have  a 
right  to  supply  it  either  as  -^0  or  — 0.  Suppose  the  order 
of  signs  to  be: 

-|-  -P  —    0 1 ,  and  for  0  writing  -|-  or  — , 

we  have :   -| — | i j and 

_!__!_ 4_ . 

In  the  upper  line  are  5  variations,  2  permanences. 
In  the  lower,  3  variations,  4  permanencies. 

Now,  if  all  the  roots  are  supposed  to  be  real,  there  will 
by  the  first  arrangement  be  5  positive  roots  and  2  negative; 
by  the  second,  3  j)ositive  and  4  negative.  There  are,  then, 
two  roots  which  have  changed  about,  being  in  one  case 
positive  and  in  the  other  negative.  But  both  suppositions 
being  legitimate,  we  have  two  real  roots,  which  are  both 
positive  and  negative,  which  absurdity  shows  them  to  be 
imaginary.  Where  the  terms  between  which  the  zero  term 
is  found  have  contrary  signs,  we  can  predicate  nothing 
about  the  nature  of  the  roots,  because  in  that  case  the 
number  of  variations  and  permanences  will  be  the  same, 
whether  we  suppose  the  absent  term  to  be  positive  or  neg- 
ative. 

EXAMPLES. 

How  many  imaginary  roots  in — 


30  PRINCIPLES  OF  ALCtEBEA, 


CHAPTER     IV. 

Transformation  of  Equations. 

Art.  35.  The  changing  the  form  of  an  equation,  au J 
yet  preserving  the  equation  is  an  operation  not  only  allow- 
able but  often  of   the  greatest  convenience. 

We  have  seen  already  (Art.  31),  that  the  signs  of  the 
toots  of  an  equation  may  all  be  changed  by  changing  the 
signs  of  the  alternate  terms;  that  is,  the  changing  of  the 
signs  of  the  terms  in  this  manner  gives  another  equation 
whose  roots  are  numerically  the  same,  but  have  opposite 
signs  to  those  of  the  first  equation. 

Art.  36.  To  transform  an  equation  into  another  equatiov 
whose  roofs  shall  be  some  multiple  of  the  r^oots  of  the  first . 

Let  the  equation  be  x""  +  Px"""^  Qx'"-'^ f  Tx^ 

?7=  0, .  .  .  .(1)  and  suppose  its  roots  to  be  a,  b,  c,  etc.  It 
is  required  to  produce  an  equation  of  which  the  roots  shall 
be  ka,  kb'kc,  etc, 

y 
Make?/=A'.r  .  '  -x^^-,  and  this,  substituted  for  x,  gives 

'•'^.  +  Pp^.+  Q^JZ+ +4+  ^='^'  ■■■■  <^'- 

Multiplying  by  t",  we  get: 

y'-J^Pky'--'-{-  Qkfif"-^^  .  .  .  .  -L  Tt"-'yJ^  Uk-"  -=  0 (8). 

In  this  equation,  since  if=^kx,  the  roots  are  ka,  kb,  kc, 
etc. 

Art.  37.  This  transformation  leads  to  one  of  the  most 
important,  which  is 

To  clear  an  equation  of  fractions  and  yet  keep  the  co-effi- 
cient of  the  highest  power  unity . 

F  H 

Let  x'"^~x'""^Qx"'-'-^~x"'-^-^  etc.,  =  0  be  an  equa- 
fc  (J 

tion  having  fractional  co-efficients,  and  j)lace  y=gkx,  that 
is,  equal  to  x  multiplied  by  the  least  common  multiple  of 
the  denominators.     Then 


TRANSFORMATION    OF    EQUATIONS.  31 

jr    ,  P.V--       Qy"-        By'"-'    ,  ^^^   _o 

Multiply  by  ^""Y/"',  and  we  have 

ir-\-gF]r-'-{^kYQir-'-{'FcfRy'''-'  -f-  etc.,  =0. 

This  is  an  equation  of  the  reduced  form,  wherein  (if  the 
roots  of  the  original  equation  are  a,  b,  c,  etc.)  the  roots  are 
kga,  kgb,  hjc,  etc. 

If  the  denominators  are  numbers  we  may  obtain  a  trans- 
formed equation  of  greater  convenience  by  assuming  for 
Ay/ a  number  less  than  the  L.C.M.  of  the  denominators, 
but  which  shall  be  such  a  product  of  prime  factors  of  the 
denominators  as  shall  secure,  after  the  substitution,  an  en- 
tire quotient  in  each  co-eflficient.  This  will  be  a  matter  of 
inspection  and  discretion  to  be  used  in  each  example. 

For  instance  take  the  equation 

-*--|^+lV~i-9M)  -  «■  ^''^^  9000istheL.C.M. 
of  the  denominators,  but  the  3d  and  4th  powers  of  9000 
are  inconveniently  larger . 

But  0  =  2x3;  12  =  2=x3;  150=2x3x5%  and  9000  =- 
2^'X3'^X5^ 

Suppose  that  in  the  example  we  make  y  =  2x3x5'^  = 
30.r;   we  shall  obtain : 

y'      _       ^f       .        W      _    _Jy^  _  _  13  _ 
2%3\5^       6.  2^3^  5=^"^  12.2^3^5^      150.2.3.5      9000       ' 
Now',  the  denominator  9000  =  2^  3'.  5=*  is  the  most  diffi- 
cult one  to  provide  for,  and  yet  it  will  disappear  when  we 
multiply  by  2\  3*.  5*.     The  result  will  be : 

,;*_5. 5,y3^5. 3. 5y_7. 213^5//— 13.2.3^5  =  0;    or, 
■,/^_25?/^+375i/'^— 1260^— 1170  =  0. 

If  the  roots  of  this  equation  can  be  found,  those  of  the 
first  will  result  from  the  relation  y=  30.r. 

EXAMPLE  2. 

7        11       25 
,r^—  ^.r"'-[-— .r— ^r,=0 .     If  we  make  ?/  =  2  X  3r,  we  get  if— 
6         Ob       72 

14?/'^-fll?/— 75=0. 


32  /  .  PRINCIPLES  OF  ALGEBRA, 

EXAMPLE  3. 

,      13  ,     21  3      32    .,     43^        1       ^ 

12     ^40        225        600      800 

12  =  2^3 

40  =  2^5 

225  =  3^5^ 

600  =  2^3.5^ 

800  -=  2^5^      The  prime  factors  being  2,  3  aud  5  ifc 

might  appear  that  2.3.5  would  be  a  proper  multiplier  for 

X,    but    on    trial    we    would    find     at     the    third    term 

21.2^3^5^/      21.3^5^^         •      ;,      n    ^      .•         x>  .  -^ 
o3  f^  o3  Q3  K3  ^^ o —    '  ^^  irreducible  traction.    But  ii  we 

Jt.O.ji  .0.0  2i 

use  the  product  2^3.5  we  shall  obtain 

,/_ 65?/*+  1,8902/^—30,7201/2—  928,800?/—  972,000  =  0   in 

which  X  =  ^^  V . 
60  ^ 

If  the  equation  has  the  co-efficient  of  the  highest  isomer  dif- 
ferent froyn  unity;  divide  through  by  that  co-efficient  and 
then  proceed  as  before . 

Suppose  3?/^~5?/-|-r=0,  or  i/^-i- -^/^—^^/^-^  =  0 . 
Put..r=3?/.  •  .^  -^+  l  =  ^'^'-  ^ -35-^+ryr=:0. 

Art.  33.  From  equation  3,  Art.  36,  we  see  that  if  the 
second  term  of  an  equation  is  exactly  divisible  by  k,  the 
third  term  by  F,  the  fourth  by  P,  etc.,  its  roots  will  have  a 
common  divisor  k. 

And  any  equation  may  be  transformed  into  another  of 

which  the  roots  are  -  of  those  of  the  former  by  dividing  the 

second  term  by /:;,  the  third  by  P,   the  fourth  by  k^,  etc. 
This  would  give  at  once  the  result  of  making  the  multiplier 

7  instead  of  k  in  the  transformation  of  Art.  36. 
k 

For  an  example  take  the  equation 
.x^ — 8^^ — 5^-4-84=0.  .  .(1)  and  let  the  second  co-efficient  be 
divided  by  2,  and  the  succeeding  co-efficients  respectively 


TRANSFORMATION    OF    EQUATIONS,  3li 

5  84 

1)V  4  and  8.     We  i^et  y'^lr'—-x  -[-  ^  '=  0 .  .  .  .(2) .     The 

4  o 

7      3 
roots  of  equation  (1)  are  7,  — 3  and  4,  and  ■^, — -  and  2  will 

verify  eq.  (2). 

Art.    39.      To  trniixform   an  equation  into  anotlier,   tlie_ 
roots  of  which  shall  be  the  reciprocals  of  those  of  the  first. 

Substitute      for  .r  in  the  equation  ' 

y 

x"'-^Px"'-'-{'Q.r"'-'^ +  2:r+C7=0,    and   the   result  is 

^^.  +  ^_,  +  ^.+ +^+  t7=0;   whence,    by 

clearing  and  reversing  the  order  of  the  terms  and  dividing 

by  ^.  y  +  ~y"-  +  §r-'+.  ..+^y'  +  ^y+\j=o. 

in  which,  if  the  original  roots  were  a,  6,  c,  these  roots  are 

\'  \'  l  '^- 

Art.  40.  If  any  term  is  wanting  in  the  given  equation, 
there  will  be  one  wanting  in  the  transformed  equation  at  the 
same  distance  from  the  last  as  the  other  was  from  the  first 
term.  If  the  original  equation  is  wanting  in  the  second 
term,  the  one  next  to  the  last  will  be  wanting  in  the  trans- 
formed one,  because  the  latter  coeJB&cients  are  equal  to  the 

former  divided  b}^  C7,  and-=r=r  0. 

Art.    41.     If   an   equation   be  transformed   by  making 

a:  =  --,  and  the  transformed  equation  should  have  the  coeffi- 

V 
cients  identical  with  those  of  the  given  equation,  but  in  re- 
versed order,  the  two  equations  are  one  and  the  same. 
This  is  evident  upon  sight,  and  therefore  their  root^  must  be 
the  same.  If  the  roots  of  the  original  equation  were  a,  b, 
c,  d,  etc.,  the  roots  of  the  transformed  one  must  also  be  a, 

5 


84  PRINCIPLES    OF    ALGEBRA. 

h,  c,  etc.     But  we  know  that   the    roots  are  also      ,    r,    -, 

a     b'    (' 

1  1  T 

etc.:     hence    the    roots    of    both   are:    a,   -      b 


«'      '     b> 


c. 


c 


etc.  ai^—px^-\-qx^—px-\-l  =  0,  ./-^-f  (/j^'+l  =  0,  .r*-f  1  =0 
in  which  the  coefficients  are:  1  — j^  +?  — P  +!»  1  -i-q  -fl* 

1  -]-lj  ^^^  of  the  kind  whose  roots  are  of  the  form  a  and  -. 

a 

Art.  42.  If  we  have  an  equation  of  an  odd  degree,  or 
one  of  an  even  degree  without  its  middle  term,  and  the 
signs  of  the  corresponding  terms,  counting  from  first  to 
last,  and  from  last  to  first,  are  opposite,  the  roots  will  also 

be  of  the  form  a,  -.    Because,  if  we  obtain  the  transformed 
a 

equation,  and  then  change  the  signs  throughout,  we  do  not 
affect  the  roots  at  all  (Art.  31),  and  yet  it  becomes  identi- 
cal with  the  original  equation,  and  must  therefore  have  the 
same  roots.     For  example,  let 

j^-^px'^-{-px — 1  =  0. 

Substituting  -  for  x,  we  have,  after  clearing, 

^—Pl/+py^—/  =■  0;  or,  y^—py'^py—1  =  0. 

Equations  whose  roots  are  of  the  form  a,  ,  b,  ,  etc., 
are  called  recurring  equations. 

Art.  43.  A  7'ecurring  equation  of  an  odd  degree  must 
have  1  for  a  root  when  the  absolute  term  is  — 1,  and  — 1 
for  a  root  whe'u  the  absolute  term  is  -f  1;  because  these 
numbers  being  substituted  for  x  will  satisfy  it. 

j^et  o(f — px^-\-qo(^^xi^'^px — 1  =  0,  and  substitute  -fl  for 
.r,  we  get  1 — p-^q — q^p — 1  =^  0.     The   other   roots  (Art. 

42)  will  be  of  the  form  a,  -,    6,  -  ,  etc. 

Art.  44.  To  transform  an  equation  into  another  of 
which  the  roots  shall  be  the  squares  of  the  roots  of  the  first. 


TRANS  FORMATION    OF    EQUATIONS.  35 

Let  us  assume,  for  convenience,  that  in  the  first  member 
of  the  equation  the  even  numbered  terms  are  negative,  and 
transpose  all  the  negative  terms  to  the  second  member. 
We  shall  have: 

x"'-[-q.r"'-^'~\'ii.r"'~^^  etc.,  =i px"'~' -\-r.r'"'^ -\-  etc. 
Square  both  members,  and  we  have: 

.r^'"+ 2c/,/-""-^  -f  (^'H- 2«).r^'"-^-|-  etc.  ,= jo V^^-^-j- 'Ipraf'"- ^+  etc. 
And  therefore  ,](f"'^{1q—p'yr""-''^(q'-\-2s—'lpr)x""-*^  etc., 
=  0.  Now  this  is  a  true  equation,  as  we  have  a  right  to 
square  both  members.  Let  y  =  x^,  and  substitute  in  the 
last  equation;  the  result  is:  y"'^{^q — p^)y'"~^-\-{(f-^'l>i 
—2pr)y'"~--\-  etc.,  =  0,  an  equation  whose  roots  are  the 
squares  of  those  of  the  first. 

EXAMPLE. 

Let  x^-^-Sx'^ — 6x — ^8  ^  0.  In  this  by  transposition  we 
have  x^ — Qx  =  8 — 3,^"^,  and  by  squaring,  x^ — 12x*^36x^  rz= 
dx* — 48a7'^-|-64;  whence  x^ — 21j7*-|-84-r^— 64  =  0,  and  placing 
y  =  x\  2/^— 21//'^-]- 84?/ — 64  ==0.  By  trial  we  find  — 1  is  a 
root  of  the  given  equation,  and  "dividing  it  out,"  we  find 
the  others  to  be  —4  and  2.  Squaring  these,  we  get  1,  16 
and  4,  which  are  roots  of  the  new  equation  and  will  verify 
it. 

Art.  45.  To  transfoi^m  an  equation  into  anotfier  whom 
roots  shall  be  greater  or  less  than  those  of  the  first  equation  by 
any  given  quantity. 

First  place,  if  necessary,  the  equation  in  the  reduced 
form: 

^'«  _j_  Px"''-'-{-  g.^'«-^-]- +  Tx^  U=0, (1) 

Let  x^  be  any  given  quantity,  and  makey±x'=x.  The  new 
equation  in  y  will  have  its  roots  greater  or  less  than  the 
roots  of  the  original  one  by  x\  Let  us  use  the  -f  sign 
only;  the  results  of  substituting  y — x^  would  only  differ  in 
sign  at  the  appropriate  places.  Substituting  y-f-x'  for  x, 
we  obtain: 
{y+x'r^P(y-\-xT-'-{-Q{y+^r'-'^--  +  T{y^x')+U=0. 

KoW)  develop  by  the  Binomial  Theorem,  and  arrange  ac- 


m 


PRINCIPLES  OF  ALGEBRA. 


cording  to  the  a. trending  powers  of  the  unknown  qiianfifi/  ij, 
(which  is  done  merely  for  subsequent  convenience),  there 
will  result: 

+     +     -f      -h     +     + 


-    H^ 

s 

^ 

•* 

§ 

■» 

i 

f 

■^o 

+  +  + 

+ 

-U 

^ 

§' 

jo 

T 

+ 

^ 

i—i 

Ss 

s 

s 

5: 

> 

> 

i 

i 

1^ 

+ 


fcO 


+ 


11 


I 

+ 


+ 


+ 


And  this  is  the  required  transformed  equation,  but  with 
the  usual  order  of  the  terms  reversed.    If  P\  C/ ,  etc.,  rep- 


TRANSFORMATION    OF    EQUATIONS.  37 

resent  the  values  of  the  co-efficients  from  y'""'  down,  and 
the  usual  descending  order  be  resumed,  the  equation  will 
be: 

y"'-\-P']r-'+Q'y"'-'^ . .  •  •+2"'2/+  6^  =0, ...  .(3) 

Art.  46.  A  method  of  arriving  at  the  values  of  the 
transformed  co-efficients,  P' ,  Q',  E\  etc.,  which  is  preferred 
by  some  as  being  shorter,  is  as  follows: 

Divide  the  first  member  of  the  equation  to  be  transformed  by 
X  minus  the  differ enee  betiveen  the  old  and  new  roots,  the  re- 
mainder ivilt  be  tJie  new  absolute  term;  divide,  by  the  same, 
this  quotient,  and  the  remainder  icill  be  the  co-efficient  of  the 
first  power  of  the  new  unknown;  divide,  by  the  same,  the  last 
quotient  obtained  and  the  remainder  will  be  the  next  co-effi- 
cient in  order,  and  so  on  to  the  last  co-effi<iient. 

Equation  (3)  of  the  preceding  article  was  obtained  from 
eq.  (1)  by  making  x  --=  y-\-x' ;  consequently  if  we  make  in 
(3)  y  =ix—x'  we  shall  simply  go  back  to  (1),  and  the  first 
members  will  give  an  identical  equation:  {x — x^)'^-\-P\x — 
./r')-^4- . .  ._^r'(.r— .^')-f  U'=x"'^Fx'''"'-\-Qx'''"'-\- .  .  .^Tx-\r 
U=0. 

If  we  divide  both  members  of  this  equation  by  any  quan- 
tity the  quotients  must  be  the  same;  dividing  the  fiist  mem- 
ber by  x—x\  the  quotient  is{x—x'Y""'^^P'{-^—r' )'""■'+ etc., 
and  the  remainder  U\'  which  is  the  absolute  term  of  the 
transformed  equation.  Consequently,  had  we  divided  the 
second  number,  which  is  the  first  member  of  the  original 
equation  by  x — x\  we  would  have  obtained  a  quotient  and 
remainder  identical  with  these. 

Dividing  this  quotient  by  x — .r'  we  have  as  a  remainder 
T\  and  would  have  had  it  had  we  divided  the'  quotient  in 
the  division  of  the  original  first  member,  and  so  on  succes- 
sively we  would  obtain  all  the  co-efficients  of  the  trans- 
formed equation. 

EXAMPLE. 

Transform  the  equation  x^ — 2./-^-i-3.r — 4=0  into  one  of 
which  the  roots  shall  be  less  by  1.7. 


38  PRINCIPLES    OF    ALGEBRA, 

First  Operation. 
.X.3  _  2x^  ^  Sx  —  4:    I  x—1 . 7 

x'  —  l.  Ix'  T^r^^O .  3,r4-  2.49 


2.49^—4 
2.49^—4.233 

.233  =  absolute  term  of  new  equa- 
tion, 

2d  Operation. 

.?r^— 0.3j?-f-2.49.|  x-^l.l 
jc'—l.lx  I  x-^lA 

1.4r+2.49 

1.4.r— 2.38 

-|-4.87  =  co-efficient  of  y  in  new  equation. 

3d  Operation. 
x-\-lA  I  x—lJ 


x—1,1     I 


3.1=  co-efficient  of  y""  in  new  equation. 
Hence  the   transformed  equation   is   y^ 4-3.1  ?/^-f- 4.87 */-[- 
.2:33=-0. 

These  three  divisions  may  be  performed  more  expedi- 
tiously by 

Synthetical  DivisigN. 

Art.    47.     Synthetical  Division  is  a  short  mode  of  di- 
viding polynomials  wherein  we  make  use   of   the   co-effici- 
ents  only.     Let  us  perform  the  three  divisions  above  in 
this  manner. 
1—2+3—4  I  1—1.7 

1—1.7  I  1—0.3+2.49  I  1—1.7 

_0.3+3— 4  1—1.7  I  1+1.4    ri--1.7 

— 0.3+.51  1.4+2.49       1—1.7        |"T 

2.49—4  1.4—2.38       +0 

2.49—4.233  +4.87 

+  .233 
The    successive    remainders    are   +0.233,    +4.87    and 


TRANSFORMATION  OF  EQUATIONS.  39 

-[-3.1  which  are  the  co-efficients  of  y\  y,  y^  respect- 
ively, and  hence  the  transformed  equation  is  ?/^-f  3.1  y'^ 
4_4.87?/^0.233=0. 

In  this  skeleton  division  we  have  simply  applied  the  rule 
for  division  of  polynomials.  Since  the  first  term  of  the 
divisor  is  unity,  the  first  term  of  every  quotient  must  he  the 
same  as  the  first  term  of  the  dividend. 

To  get  the  second  term  of  the  quotient  we  have  multi- 
plied the  second  term  of  the  divisor  by  the  first  term  of  the 
quotient  and  then  subtracted  this  result  from  the  second  term 
of  the  dividend.  And  this  difference  divided  by  1  (the 
first  term  of  the  divisor)  would  give  itself  as  the  second 
term  of  the  quotient.  Now  we  would  have  arrived  at  the 
same  result  if  we  had  at  once  changed  the  sign  of  the  sec- 
ond term  of  the  divisor,  multiplied  this  by  the  first  term 
of  the  quotient  (the  same  as  the  first  term  of  the  dividend) 
and  had  added,  this  product  to  the  second  term  of  the  divi- 
dend. 

Then  this  algebraic  sum  would  have  been  the  second 
term  of  the  quotient. 

•  The  third  would,  in  a  similar  way,  have  been  obtained 
by  changing  the  sign  of  the  second  term  of  the  divisor, 
multiplying  it  by  the  second  term  of  the  quotient  and  add- 
ing the  product  to  the  third  term  of  the  dividend.  And  so 
proceeding  until  the  first  remainder  was  found,  which 
would  give  the  absolute  term.  In  a  precisely  similar  man- 
ner an  expeditious  division  may  be  made  of  the  succes- 
sive quotients  until  all  the  co-efficients  of  the  transformed 
equation  shall  have  been  obtained . 

It  will  be  observed  that  these  operations  are  performed 
without  making  any  use  of  unity  as  the  first  term  of  the  di- 
visor. We  have  then  for  the  synthetical  division  used  in 
this  transformation  the  following 

RULE. 

Write  the  co-efficients  of  the  first  member  of  the  given  equa- 
tion^ ill  the  reduced  form,  in  thier  order  with  their  prefer 


40  PRINCIPLES  OF  ALGEBRA. 

signs.     Change  the  sign  of  the  quantity  which  is  the  difefence 
between  the  old  and  new  roots  and  call  it  the  multiplier. 

Ihe  first  term  of  the  dividend  is  the  first  term  of  the  quo- 
tient. Multiply  it  by  the  multiplier  and  add  fhej^roduct  to  the 
second  term  of  the  dividend,  the  result  will  be  the  second  term 
of  the  quotient;  then  multiply  this  by  the  multiplier  and  add 
to  the  third  term  of  the  dividend,  and  the  result  will  be  the 
third  term  of  the  quotient.  So  proceed  until  the  first  remain- 
der is  obtained.     It  will  be  the  absolute  term  required. 

Using  the  same  multiplier,  treat  in  the  same  luay  the  suc- 
cessive quotients  and  obtain  b^  the  successive  remainder's,  all 
the  coejfficients  up  to  that  of  the  power  next  to  the  highest 
of  the  unknown  quantity.     The  first  coefficient  is  unity. 

Let  us  apply  this  rule  to  to  the  last  example. 
1    _2      -j-3       —4:         I  +1.7 


1.7  —0.51  +4.233 

1st  quotient,  1    —0.3  +2.49,4-0.233    ' 
1.7  +2.38 

•    1st  remainder. 

2d  quotient,    1    +1.4,+4.87 
1.7 

2d  remainder. 

3d  quotient,    1,  +3.1  3d  remainder. 

And  the  equationas  before,  is  ^'+3.1?/'+ 4. 87?/+ 0.233=0. 

Art.  48.  This  discussion  of  Synthetical  Division  is 
here  adduced  for  the  information  of  the  student,  and  to  be 
used  in  similar  transformations,  and  especially  in  the  op- 
erations of  Horner's  Method  of  approximating  to  the  roots 
of  numerical  equations.  But  it  is  thought  best  for  him  to 
defer  its  use  until  he  is  familiar  with  the  principles  of  the 
main  operations  which  it  is  intended  to  facilitate. 

Thus  it  would,  on  first  going  over  this  subject,  probably 
be  better  for  the  student  to  obtain  the  various  coefficients 
of  the  transformed  equation  by  the  ordinaiy  division  of 
polynomials. 

In  fact,  where  the  numbers  are  not  large  and  do  not 
have  to  be  raised  to  very  high  powers  it  will  be  as  well  to 
miike  the  plain  and  simple  substitutions  and  to  perform 
the  indicated  operations,  as  follows: 


TKANSrOIlMATION  OF  EQUATIONS.  41 

Resuming  the  same  example :  to  find  an  equation  whose 
roots  are  1 . 7  less  than  those  of  the  equation 

.r3_2jr2-f  3.^—4  =  0. 
Place  j-^^jz+l.T         (1.7)^=2.89;  (1.7f=4.913 

.r'  =  {y^l.7f==if  +  5.1//  +  8.07//  -f  4.918 
—2x'=  —2(ij[-l.lf=  —  %f  —  6.80//  —  5.780 
4-3.r  =      3(//+1.7)  =  -I-  3.00//  +  5.100 

—  4  =         ^ — _4 

//'+3.1/     +  4.87//  4-  0.233  =  0 

EXAMPLES. 

1.  Find  an  equation  whose  roots  are  less  by  1  than 
those  of  j^— 7a-+7=0.     Ans.  //'+3//»— 4//H-l=0. 

2.  Find  the  equation  whose  roots  are  greater  by  p.  than 
those  of  x^ — 2>a;=-f  (/.r — r=0. 

Ans .   //^— (3^+p)//'-f (3f?'''— 2/x^+f7)//— ((^  fjx^-^— ^^+r)=0 . 

3.  Find  the  equation  whose  roots  are  greater  by  2  than 
those  of  ;r*--2.r^+5.r^+4a:— 8=0.  Ans.  if—lOy'-^^lif— 
72//+36=0. 

4.  Find  the  equation  whose  roots  are  greater  by  1  than 
those  of  x^— 5.r'~0.r— 2=0.     Ans.  //*— 4//'+//"'=0. 

As  //  is  twice  a  factor  of  every  term  of  this  transformed 
equation,  let  us  "divide  out"  //^,  and  we  have  .//^ — 4y-j-l 
=  0,  whose  two  roots  are  2-\-\/3  and  2 — yS,  and  as  the 
four  roots  are  0,  0,  2-l-i/3,  2 — 1/3,  if  we  subtract  1  from 
each  we  get  the  roots  of  the  given  equation,  — 1,  — 1,  l-j- 
l   3,  1-,   3. 

Art.  49.  2h  transform  an  equation  into  another  ivantbig 
the  second  or  any  i^ctrticular  term . 

From  equation  (2)  of  Art.  45  we  see  that  the  coefficient 
of  y"'-'  (which  is  the  second  term  in  the  usual  arrangement) 
is  mx'-\-P.     Now,  since   x'  is  entirely  arbitrary,  we   can 

P 

give  it  such  a  value  that  mx'-\-P  =^0,  .'.  x' = ;   that 

m 

4 


42  PRINCIPLES    OF    APGEERA. 

is,  minus  the  coefficient  of  the  second  term  divided  by  the 

exponent  which  denotes  the  degree  of  the  equation  to  be 

transformecl.     All  we  have  to  do  is  to  substitute  for  x  the 

P 
quantity  y .     The  equation  resulting  will   have   roots 

P 
greater  by  —  than  the  original  roots. 
m 

To  cause  the  third  term  to  be  absent  from  the  new  equa- 
tion,   we   must   place   the   coefficient    of    ;v'"~',  which    is 

~^-^-  ,2''^-j-(m — l)P.r'-l  Q,  =  0,  and  solve  this  quadratic 

to  get  the  requisite  value  of  .t\ 

To  cause  the  fourth  term  to  be  absent  it  will  be  neces- 
sary to  solve  an  equation  of  the  third  degree;  the  next 
coefficient  would  give  an  equation  of  the  fourth  degree, 
and  so  on  upward.  These  equations  would  be  difficult  or 
impossible  to  solve . 

EXAMPLES . 

1.  Transform  cr^ — 6x'^-f7  =  0  into  an  equation  where 
the  second  term  is  absent.  Ans.   y^. — 12y — 9  =  0. 

2.  Transform  x* — 8.r'— 5j--[-12  =  0  into  an  equation 
whose  second  term  is  wanting. 

Ans.  y'—24:y''—Gdy—4:G  =  0. 

It  sometimes  happens  that  the  same  value  of  x'  will  sat- 
isfy both  the  equations  arising  from  jDutting  the  coefficients 
of  the  second  and  third  terms  =  0.  In  this  case  those  two 
terms  will  vanish  simultaneously. 

As  an  example :     ,x^-|- 4./'^-J- (jx^-\-3x^ 4  =  0. 

p      4 

Here ^=  -r-  =  — 1  =  x\  and  substituting   y-\-x'  = 

m  4 

y — 1  for  ,x:  . 

y^-^fJr  6/-  42/+1 

4:y^—12y'-\-12y—4: 

Gf-12y-\-Q 

3?/-3 

+4 

2/*  — 2^-j-4  =^0  is  the  transformed  equation . 


TRANSFORMATION    OF    EQUATIONS.  43 

Or  by  successive  divisions: 


n.if-\-'Sx  2;r'^-|-2.r  x-^1     .r+1  1 


3j;''-|-3ic  X  jfl  ,0 

,-1 

Giving  the  remainders  4,  — 1,  0,  0,  and  consequently  the 
equation   ^' — ^-{-4  =  0. 

The  same  example  by  Synthetic  Division  is  as  follows: 
1  +4  +6  +3  +4  I  -1 
—1  —3  —3  —0 


1  -1-3  -j-3  -}-0,-j-4 
—1  —2  - 1 

1  -h2  -Tl,^! 
—1—1 


1  -f  1,-f-O 
—1 


i,H-0 

Here  the  remainders  are  as  before,  and  the  e({uation  is 

DERIVED    POLYNOMIALS. 

Art.  50.  By  examining  equation  (2)  Art.  45,  we  see 
that  the  coefficient  of  y"  is  simply  the  first  member  of  the 
equation  which  was  transformed  with  a  dash  placed  on  x. 
Omit  the  dash,  and  let  this  be  denoted  by  f(x). 

The  coefficient  of  i/  is  formed  ffom  f{x)  by  multiplying 
each  coefficient  by  the  exponent  of  ./■  in  the  term  and  di- 
minishing that  exponent  by  unit3^  Let  this  coefficient  be 
denoted  by/'(.x). 


44  PRINCIPLES    OF    ALGEBRA. 

The  numerator  of  the  coefficient  of  if  is  formed  from 
f\-r)  by  the  same  law  as  that  by  which  f\-r)  av.is  derived 
fromy(;r).    The  denominator  is  2  or  1.2.  Let  it  be  denoted 

by /_::«. 

•^     1.2 

And  the  law  by  which  any  of  these  coefficients  is  ilerived 
from  its  immediate  predecessor  is: 

.Multiply  each  term  of  the  preceding  coejficient  by  the  expo- 
nent of  X  in  that  term,  diminuih  this  exponent  by  unity  and 
divide  the  alyebraic  sum  of  the  results  by  the  number  of  pre- 
ceding coefficients. 

These  coefficients,  after  the  tirst,  are  derived  from  their 
immediate  predecessors  by  the  same  law  as  that  by  which 
the  coefficients  of  the  Binomial  Formula  are  built  \\]}. 

The  student  of  Calculus  will  recognize  the  numerators 
as  differential  coefficients  of  the  iirst,  second,  third  order, 
etc.  They  are  called  derived  functions,  or  derived  polyno- 
mials. 

Thus  f^{x)  is  the  Jirst  derived  polynomial  off{x). 
/"(.r)  is  the  second  deriiied  polynomial . 
f^^'{x)  is  the  third  derived  poly )iomial . 
etc.,  etc.,  etc. 

Mark  the  distinction  between  the  derived  polynomials 
and  the  coefficients  of  tlie  cleveJopment  in  equation  (2)  Kvi. 
45.  The  tirst  coefficient  is  the  original  first  member  with 
x^  in  place  of  x;  the  second  coefficient  is  the  first  derived 
polynomial;   the  third  coefficient  is  \  of  the  second  derived 

polynomial;   the  fourtli  coefficient,      ov  .^  o  of   the  third   de- 
rived })olynoniial,  etc. 

EXAMPLES. 

Let  'dx'-\-QiX^—Zx'-^'2x-\-l  =  0. 

f\x)  =  12^+18:c'^— 6a'+2. 
/"(jc)  =  36£c^4-36.r— 6. 
/•'"(ic)  =  72.T-f  36. 
/--(«.)  =  72. 


^-  ^  ^ — JL  —  0;    whence    clearing"    of 


TRANSFORM  VnON    OF    EQUATIONS.  45 

The  last  terms  having  .if,  the  terms  into  wliich  0  is  mul- 
tiplied do  not  appear  in  the  sacceediny;  derived  polynomi- 
als. In  this  way  1,  2,  —6,  3()  and  72  are  successi  'ely 
dropped,  which  terminates  the  series. 

Let  it  be  required  to  transform  the  equation 

3./-^'  I  15.r^-l  25./'— 3  =  0 
into  one  wanting-  the  second  term.     First  placing  it  in   rhe 
25 
3 
fractions  (Art.  37), 

v/M-15//-h75v/-27  =  0. 

— P 

J{-'')  ^=^  //!4-i%''-i-'^^// — '^'*  ^^"^^  -^^  =  —  —  =  — ^5 

.  • .    /(.r)  =  —125  [-375—375—27  =  —152 
/'(.r)  ^  3v'4  3()v+75;  /'(.r)  .^  75— 150 -j  75  =  0 

^T)  =  %+15;  q^^ -15+15^0 

r '(')__.  /'"W-i_i 

2.3     "  2:3     ~ 

Hence  the  equation  is  k^ — 152  ■■=  0,  an  equation  wanting 
both  the  second  and  third  terms. 

2.  Transform  .r' — 10y-[-7,r^-[-4.r — 1)  =:  0  into  an  ecpia- 
tion  wanting  the  second  term. 

Ans.     <y^— 33dt^— 118?/'^— 152//— 73  =  0. 

3.  Transform  3./"* — 13.r^4-7.r^ — 8.r — 9  ^=  0  into  an  equa- 
tion the  roots  of  which  shall  not  be  so  great  by  the  quan- 
tity J.  Ans.    ^iu'—da'~Au'—^^a--^^  =  0. 


RELATIONS  OF  THE  DERIVED  POLYNOMIALS  TO  THE  ROOTS  OF  AN 
EQUATION EQUAL  ROOTS. 

Art.    51.     Let  a,  b,  c,  d,.  .  .  .1  he  the  ni roots  of 
Then  we  know  that 


46  PRINCIPLES  OF  ALGEBRA. 

..(..'-O. 

Let  this  be  transformed  by  substituting  u-\-x  for  x\- 

.  • .  -(u^oc—I)  =  \itJ^[.x—a)lu^(x—h)]\ii^{x—r)] .... 
WM^-~i)\ (1) 

Now,  if  we  regard  (.r — a),  {x — b),  (.r — e),  etc.,  as  single 
quantities,  the  factors  of  the  second  or  third  member  of 
equation  (1)  will  be  in  the  form  of  the  continued  product 
of  the  binomial  factors  of  the  tirst  degree  which  belong  to 
the  roots  of  an  equation.  In  this  equation  the  unknown 
quantity  is  u . 

If  the  indicated  oj^erations  be  performed  in  the  first  and 
last  members  of  equation  (1),  we  shall  have  from  the  first 
member: 

f(.r)  being  the  first  member  of  the  equation  with  which  we 
started,  but  without  putting  on  the  dash  ('),  and  /'{"'), 
f'\-r),  etc  ,  being  the  derived  polynomials.  This  expres- 
sion placed  opposite  to  what  the  second  member  becomes 
is  an  identical  equation,  to  which  the  principle  of  Indeter- 
minates'  Coefficients  applies.  The  coefficient  of  W  in  this 
result  will  be  the  continued  product  of  the  second  terms 
(to  wit,  (.r — a),  {x — b),  {x — c),  etc.,  regarded  as  single 
terms)  of  the  binomial  factors  of  the  first  degree  with  re- 
spect to  u  (Art.  20),  and  the  coefficient  of  il°  in  the  first 
member  being  f{x), 

f(x)  =.  {x—a){x—b)(x—c) ....  (./'—/), 

which  we  already  knew.  But  the  coefficient  of  a  in  the 
second  member  being  the  sum  of  the  combinations  of  the 
111  second  terms  (or  factors,  (.r— a),  {x — /;),  etc.)  in  groups 
of  m — 1,  we  have: 

-^  ^  '       X — a^ X — 6  '  x — 0  '  y^ — I 


TRANSFORMATION    OF    EQUATIONS.  47 

because,  /"(.«)  being  the  product  of  the  ))i  terms  or  factors, 
whenever  it  is  divided  by  one  of  them,  the  quotient  is 
product  of   a  group  of   m — 1. 

Again,  equating  the  coefficients  of  u'\  we  get: 


2  (jc—aXx—b)   f  (x—a){x—c)  '  '  (x—lc)(x—l) 

and  equating  those  of  u^: 

f"'(.r)_  /(,,■)  ^  J{^) 


2.3         (.r— nX-r— 6)(,r— r)   '   {x—a)(x—h)(.t:—(l)   '    ' 


•  + 


{r-y)(.r-k)(.—l) 
and  so  on. 

Hence  we  may  announce  in  general  language  that 

The  FIRST  DERIVED  POLYNOMIAL  of  the  first  member  of  the 
reduced  equation  is  equal  to  the  algebraic  sum  of  the  quotients 
arising  from  diriding  that  first  member  successively  and  singly 
by  the  factors  of  the  first  degree  belonging  to  the  roots. 

The  SECOND  DERIVED  POLYNOMIAL  %s  cqual  to  twicc  the  alge- 
braic sum  of  the  quotients  arising  from  dividing  the  first  mem- 
ber by  the  product  of  every  group  of  two  of  the  factors  of  the 
first  degree  belonging  to  the  roots. 

The  THIRD  DERIVED  POLYNOMIAL  is  six  timcs  the  algebraic 
sum  of  similar  quotients,  but  the  divisors  are  the  products 
of  every  group  of  three;    and  so  on. 

EQUAL    ROOTS. 

Art.  52.  The  relations  just  discussed  are  very-  impor- 
tant, but  specially  interesting  because  they  lead  to  the 
discovery  of  equal  roots,  if  an  equation  has  any. 

Suppose  an  equation  has  tw^o  equal  roots,  a;   then 

/(./j)  =  (.X — a){x — a){x: — b) (x — I); 

and  the  first  derived  polynomial, 

/»-^M  +  M  +  M  +  etc., 
^  '       X — a^  X — a  '  X — b 

will  have  every  term  divisible  by  x — a,  because  the  numer- 
ator of  every  term  contains  {x — a)\  and  after  the  denomin- 


48  PRINCIPLES  OF  ALGEBRA. 

ator,  even  when  it  is  x — «,  is  divided  out,  there  is  still  a 
factor  X — a  left.  Thus  we  see  there  will  be  a  common  divi- 
sor of  the  first  member  and  its  first  derived  polynomial. 

There  might  have  been  several  roots  of  one  value  and 
several  of  another  value.  Thus,  suppose  there  are  n  roots 
a,  r  roots  h  and  .s  roots  c;  then 

f{,r)  =  {x—a)"{x—hY{,r--cy.  . .  .(./•— A') (.r—/) ; 

-^  ^^~  x—a  +  x—a  "^ '  x—h   '   x—h^' '  x~c 

^x—c^ ^  x-k   '   x—l 

Now,  the  terms  of  f\x)  where  the  denominators  are  not 
repeated  (like  r — I-  and  x — /)  will  have  as  a  factor  (./■ — «)" 
{x — hY{x — c)\  and  every  other  term  will  have  a  similar  fac- 
tor in  which  each  repeated  factor  will  have  an  exponent  at 
least  equal  to  n — 1.  Hence  the  first  member,  /'(r),  and  its 
first  derived  pol3momial,/'(j'),  will  have  a  common  divisor, 
which  is  (x — a)""''{.v — hY-'{x—cy-'.  .  And  this  wall  be  the 
H.C.D.,  because  none  of  the  factors  which  belong  to  single 
roots  can  enter  it,  since  every  such  factor  would  be  want- 
ing in  some  term  of  f'{x)  where  it  had  been  divided  out; 

f(x) 
as,  for  instance,  x — k  would  be  absent  from  -^  . 
'  X — k 

Art.  53.  To  find,  then,  the  equal  roots  which  may  be  in 
an  equation,  we  find  the  H.C.D.  between  the  first  member  and 
its  first  derived  polynomial.  If  there  is  none,  there  aire  no 
equal  roots;  but  if  there  is  one,  2)lace  it-  equal  to  0,  and  the 
roots  of  this  equation  will  be  the  equal  roots  of  the  p^^oposed 
equation. 

Call  the  H.C.D.,  i).  li  D  is  of  the  first  degree,  there 
are  two  roots  which  are  the  same . 

If  D  is  of  the  second  degree  and  of  the  form  {x — a)', 
there  are  three  roots  a;  and  if  it  is  of  the  form  (r — a){x — b), 
there  are  two  roots  a  and  two  roots  b . 


TRANSFORMATION  OF  EQUATIONS.  49 

In  general,  whatever  the  degree  of  the  equation  i>  =  0, 
each  of  its  single  roots  will  be  twice  a  root  of  the  equation 
proposed,  and  all  of  its  repeated  roots  will  appear  once 
more  frequently  in  the  proposed  equation. 

Having  found  all  the  equal  roots,  make  a  continued 
product  of  the  binomial  factors  coirresponding  to  these, 
and  divide  f{x)  by  it;  this  will  lower  or  depress  the  degree 
of  the  equation  as  many  units  as  there  are  equal  roots  and 
render  it  far  easier  to  be  solved,  and  may  even  bring  the 
depressed  equation  within  the  limits  of  those  which  we 
know  how  to  solve  directly  and  exactly. 

EXAMPLES. 

1.  Find  the  equal  roots  in  the  equation 

^7_3^6_|_9_^3_i9^4_|_27.^_33^2_|_27.^_9  ^  q  . 

The  first  derived  polynomial  is : 

And  the  H.C.D.  between  this  and  the  first  member  is: 

Placing  this  equal  to  zero,  and  finding  the  H.C.D.  be- 
tween this  first  member  and  its  first  derived  polynomial j  we 
get  X — 1.  Then  (x — 1)^  is  a  factor  of  the  first  member  of 
the  secondary  equation  and  {x — 1)^  is  a  factor  similarly  in 
the  first  member  of  the  original  equation.  There  are  now 
known  to  be  three  roots  =  1.  Dividing  {x — 1)^  out  of 
a;*— 2ar»-|-4^'^— 6,r+3  =  0,  we  have  07^+3  =  0,  and  x  = 
~V — 3.  {j(f-\-2>y  will  be  a  factor  of  the  original  first  mem- 
ber and  the  product  of  the  factors  corresponding  to  the 
equal  roots  of  the  jjroposed  equation  is  {x — lf{x^-\-^f  --= 
original  first  member. 

2.  What  are  the  equal  roots  in 

2a;*~12a;»+19x=*— 6^+9  =  0? 

The  first  derived  polynomial  is  8x^—S6x^-\-SSx — 6,  and  the 
H.C.D.  =07— 3. 


50 


PRINCIPLES  OF  ALGEBRA. 


There  are  two  roots  =  3,  and  the  others,  after  dividing 
out  and  dei^ressing  to  an  equation  of  the  second  degree, 

are  found  to  be  — ^r —  and ^ — . 

3.  Find  the  equal  and  other  roots  of  the  equation  x^-\- 
-f  2^— 12j*— 14a^-i-47j-^-f  12.r— 36  =-  0. 

Ans.   two  =  2,  two  ^=  — 3   and  also   1  and  — 1. 

4.  What  are  the  roots  of  .x^-f  4.r*— 14a?''— IT.i-— 6  =  0  ? 

Ans.    three  =  — 1,  and  besides  2  and  — 3. 


CHAPTER     VI. 

Limits  and  Places  of  Roots. 

Art.  53.  A  rational  integral  function  of  x  is  one  in 
ivhich  the  exponents  of  x  are  lohole  numbers  and  the  coefficients 
are  independent  of  x . 

Thus  fix)  =  x""  -f  Px'"-'-{-  Qx'"-'-\-  .  .  .  .-\-Tx'-\-  Vx  -f  U, 
in  which  m  is  a  positive  whole  number  (integer)  and  P,  Q, 
T,  etc.,  are  independent  of  x,  is  a  rational  integral  func- 
tion of  X  of  the  With  degree. 

Art.  54.  In  any  rational  integral  function  ofx  arranged 
according  to  the  descending  powers  of  x,  any  term  which  is 
present  may  be  made  to  contain  the  sum  of  all  ivhich  follow  it, 
as  many  times  as  we  please,  by  taking  x  large  enough. 

And  any  such  term  may  be  made  to  contain  the  sum  of  all 
ivhich  precede  it.  by  taking  x  small  enough. 

In  fix)  =  xr-\-Px'''-'-\-Qx"'-'-\-  . .  -\-Sx'"-"-^-'-\-Tx'"-*'-{- . . 
.  .  .  .-\-U,  Sx'"-"-^'  will  be  the  /«th  term,  and  may  contain 
the  sum  of  all  which  follow  it,  if  x  be  large  enough.  If  it 
can  be  made  to  contain  something  larger  than  that  sum,  it 
will,  of  course,  contain  that  sum.     Now,  suppose   all   the 


LIMITS  AND  PLACES  OF  ROOTS.  51 

terms  after  Sx'"-**-^'  to  have  the  largest  coefficient  among 

them.     Let  it  be  L;   then  X(,r"'-"-{-.r"'-''-'-j-,r"'-"-'-f -f 

j^;-{-l)  >  ^^^  sum  of   the  terms  following  the  nth,  and  = 

jr,/__ \       Divide  the  /ith  term  by  this:   L{x"'-''-^' — 1) 

X — i 
_S  ^    x'"-i'{x~l)      S  x-1 


By  increasing  x  we  may  increase  the  numerator  indefi- 
nitely, and  at  the  same  time  make  the  denominator  as  near 
unity  as  we  please .  Consequently  the  ?ith  term  will  con- 
tain those  that  follow  as  mnjij  times  as  desired.  This 
proves  the  first  j)art  of  the  proposition. 

Suppose  we  make  x  =  --,  then  increasing  t/ diminishes  .r. 

We  have:  ~['^-{-Py-{-Qif^- ....  +^2/"-^+2'r+  •  •  •  -^-WV 
The  series  within  the  brackets  is  such  that  any  term,  as 
Sy'*~',  may  be  made  to  contain  the  sum  of  all  which  pre- 
cede it,  'i^-\-Py-\-Qf/-{~  etc.,  as  often  as  we  j)lease,  by  taking 
y  large  enough,  which  means  .r  small  enough.  This  is 
evidently  shown  by  the  same  reasoning  as  in  the  first  case, 
and  establishes  the  second  branch  of  the  proposition. 

Art.  55.  The  first  term  of  the  function  may  be  made 
to  contain  the  sum  of  all  of  its  successors  any  number  of 
times. 

Art.  56.  A  variable  quantity  is  said  to  increase  or  de- 
crease under  the  law  of  continuity,  when,  in  passing  from 
one  designated  state  to  another,  it  passes  through  every 
intermediate  state  without  interruption .  A  taper  burning 
away,  a  cask  of  fluid  being  discharged  by  a  cock,  a  plant 
growing,  present  instances. 

Let  it  be  shown  that  if  x  increases  or  decreases  under 
the  law  of  continuity,  that  f{x)  will  increase  or  decrease 
under  the  same  law. 


52  PRINCIPLES  OF  ALGEBRA. 

When  .T  =  a,  let /(a)  designate  the  corresponding  state 
of  the  function;  when  x  =  h,  f{b)  the  state  of  the  function 
then  corresponding,  etc.  Suppose  that  x'  were  a  certain 
value  of  X,  and  give  it  a  small  increment,  u.  "We  see  from 
the  development  (2)  of  Art.  45,  that  we  shall  have: 

f{x^^u)  =/(^')+"A.rO+ J^  ......  .    J^  ........ 

1.2-^  ^  ''^i.2.3'^   (•^;+--- 


1.2.3.. 


f"'\x'). 


In  (2)  of  Art.  45,  ^--— f'"\x') 

\  .  Z  .  o  .  .  .Ill 


Now,  if  we  transpose  the  term  f{x')  to  the  first  member, 
we  have: 


f(x'+u)-f{.x')  =  uf'{:^')+^"{.r')- 


IL 


The  first  term  in  the  second  member  (which  is  present) 
may  be  made  indefinitely  greater  than  the  sum  of  all  which 
here  follow  it  (they  Avould  precede  it  in  the  arrangement 
of  Art.  55),  by  taking  the  increment,  n,  small  enough. 
But  when  u  is  taken  extremely  minute,  although  the  first 
term  in  the  second  member  will  contain  the  sum  of  the 
following  terms  an  indefinite  number  of  times,  the  first 
term  itself  becomes  indefinitely  small.  Hence  the  differ- 
ence between  the  states  of  the  function,  f{x'-\-u)—f{x'),  be- 
comes inappreciable.  Hence, when  the  successive  increments 
of  X  are  indefinitely  small,  and  x  varies  under  the  law  of 
continuity,  the  function  of  x  will  vary  under  the  same  law. 

LIMITS    OF    ROOTS. 

Art.  57.  The  limits  of  the  roots  of  an  equation  are 
values  between  which  all  the  roots  exist. 

A  superior  limit  of  the  positive  roots  is  any  number  or 
quantity  of  their  kind  greater  than  the  greatest  of  them. 


LIMITS  AND  PLACES  OF  ROOTS. 

An  inferior  limit  of  the  poi^itive  rootfi  is  any  number  or 
quantity  of  their  kind  less  than  the  least  of  them . 

A  superior  limit  of  the  negative  roots  is  a  number  or  quan- 
tity of  their  kind  tvhich  is  negative  but  numerically  gr-eater 
than  all  of  the  negative  roots. 

An  inferior  limit  of  the  negative  roots  is  a  number  or  quan- 
tity of  their  kind  which  is  negative  but  numerically  less  than 
any  of  the  negative  roots. 

Since  a  root  of  an  equation  whose  second  member  is 
zero,  when  substituted  for  the  unknown  quantity,  will  re- 
duce the  first  member  to  zero,  if  we  put  for  the  unknown 
quantit3'  any  quantity  greater  than  ^he  greatest  of  the  j^os- 
itive  roots,  the  first  member,  when  reduced,  will  be  found 
greater  than  zero,  that  is,  positive.  This  number  and  all 
greater  than  it,  that  is,  all  b^etween  it  and  -J- go  ,  would  be 
superior  limits  of  the  positive  roots  of  the  equation.  The 
smallest  of  such  limits  which  is  attainable  is,  of  course, 
the  one  required  for  practical  use,  as  a  general  thing. 

It  may  be  j)roved  that  the  greatest  coefficient  plus  1  is  a 
superior  limit  of  the  positive  roots;  and  even  the  greatest 
negative  coefficient  plus  1  is  such  a  limit,  and  often  a  better 
one,  because  it  may  be  smaller.     This  last  is  called 


MACLAURIN  S   LIMIT. 


In  the  equation  f{x)  =.  x"'-{-Px"'-'-{-Qaf"  "-f ■^-Tx-\- 

C/'  =  0, .  .  .  .(1),  let  the  first  term  be  positive  and  the  others 
either  positive  or  negative  as  may  happen.  Let  N  be  the 
greatest  negative  coefficient,  and  suppose  what  would  be 
the  most  unfavorable  case  which  could  happen,  that  all  the 
other  coefficients  excej^t  the  first  were  equal  to  it  and  all 
negative.  These  negative  terms  would  then  form  a  geo- 
metrical progression  with  the  ratio  x,  and  it  would  be 
necessary  only  to  put  for  x  a  value  which  would  make 

i\V— 1) 


.r'">iV(~r'""'-[-.r' 


f  ^+1)  = 


x-l 


Now,  if  in  the  inequation  «;"'<— ^-^ — r—^  we  place  a:* — 1  equal 


54  PRINCIPLES    OF    ALaEBRA. 

to  N,  or  X  =  ^Y-f  1,   we  shall  satisf}'  it,  having  .r'"  >.>:"'— 1. 

Hence  the  greatest  negative  coefficient  plus  1  i.s  a  superior 
limit  of  the  positive  roots  of  an  equation. 


Art.  58.  Since  changing  the  signs  of  the  alternate 
terms  would  make  the  positive  roots  all  negative  (and  the 
negative  all  positive)  it  is  evident  that  ttie  greatest  negative 
coefficient  of  the  transformed  equation  plus  unity  would  be  a 
superior  limit  of  the  negative  roofs  of  the  equation;  that  is, 
intrinsically  less  than  all  of  the  roots.  It  would  be  numer- 
icallj'^  greater  than  any  negative  root. 

ORDINARY  SUPERIOR   LIMIT    OF    POSITIVE    ROOTS . 

Art.  59.  When  the  first  term  is  followed  immediately 
by  one  or  more  other  positive  terms,  a  closer  limit  may 
be  obtained. 

Let  us  suppose  .r"'~"  to  be  the  power  of  .r  in  the  first  neg- 
ative term,  and  take  the  most  unfavorable  case  which  could 
happen,  that  is,  that  all  the  succeeding  terms  are  negative 
and  all  have  the  greatest  coefficient  among  them.  Let  S 
be  that  coefficient.  Then  if  we  can  make  a;'">;SV"'""-|- 
Sx"'~*'~'-\- .  . .  .-\-Sx-\-S,  it  will  be  more  than  sufficient  to 
make  the  first  member  jiositive,  because,  in  fact,  x"'  would 
be  increased  by  the  addition  of  the  other  positive  terms. 

Divide  both  members  of  the  inequation  by  x^'\  and  we 
get:  ♦ 

1<^+ A_   ,   A    , -|--'^'_    .    -^ 

There  are  n  terms  before  the  first  negative,  term  and  let  us 
suppose  X  =  \/'8-\-\\  representing  the  value  of  \fS  by  >S", 
whence  ^  ==  S"",  and  .r  =  !+*§',  the  second  member  of 
the  inequality  will  become: 

1-11 ~r  /o/    \-i\m—i     I     /.C'    1    1  Xw  *^ 


(5' _|_1)«  -r  (6^'-j-l)«-'-^  '•••••    '  (5f' +1)"'-^  '  (*S^'+1)' 


LIMITS  AND  PLACES  OF  ROOTS.  55 

We    have  here  a  geometrical  progression,  in  which  the 

j&rst  term  is  —       ^^^  with  a  ratio  .  ^,  .   ....    Its  sum,  there- 

(/^  -j-  1)  \^  ^r  J-) 

fore,  is: 

S'"  S'"  S"*  S'" 


1  -i"'^      -'- 


which  is  the  difference  between  two  proper  fractions,  and 
therefore  less  than  1,  as  was  required.  The  quantity 
l,^  S-\-l  will  consequently  make  the  first  member  of  the 
given  equation  positive,  and  be  a  superior  limit  of  the  pos- 
itive roots.  This  result  may  be  stated  in  common  language 
thus : 

Extract  that  root  of  the  greatest  negative  coefficient  of  which 
the  index  is  the  number  of  terms  before  the  first  negative  term; 
increase  this  by  1,  and  the  res\dt  will  be  a  good  superior  limit 
of  the  positive  roots -of  the  equation.  If  any  term  is  absent, 
it  must  be  counted  to  determine  the  index  of  the  root. 

If  n  =  1,  the  second  term  is  negative,  and  ^V'S-^-l  = 
S-\-l,  the  same  as  in  Art.  57. 

EXAMPLES . 

Find  a  superior  limit  of  the  positive  roots  of  x*-\-llx^ — 
2Bx—ei=  0. 

Here  n  =  3,  and  the  limit  is  ^^67-|-l.  The*  cube  root 
of  67  is  between  4  and  5,  and  hence  5-f  1  will  be  the  limit. 

2.     ^*+llx=^— 25r— 61  =  0.     Limit  =  1^61+1,  or  5. 

INFERIOR    LIMIT    OF    POSITIVE    ROOTS. 

Art.  60.     If  in  any  equation  we  make  x  =  ~,  the  roota 

y 


56  PRIJsCIPLES   or    ALGEBRA. 

of  the  transformed  equation  being  reciprocals  of  tliose  in 
the  first,  the  greatest  positive  root  of  the  transf onned  will  be 
the  reciprocal  of  the  least  positive  root  of  the  given  equation . 
Hence  to  obtain  the  inferior  limit  of  the  positive  roots: 

Substitute  -  for  x;   find  the  superior  limit  of  the  positive 

roots  of  the  transformed  equation ;  its  reciprocal  will  be 
the  limit  required. 

SUPERIOR    LIMIT   OF   NEGATIVE    ROOTS. 

Art.  61.  This,  as  already  indicated  (Art.  58),  will  be 
the  superior  limit  of  the  positive  roots  of  an  equation 
whose  roots  have  signs  opposite  to  those  of  the  given  equa- 
tion. This  transformed  equation  can  be  had  by  making 
X  =  — y,  or  by  changing  the  signs  of  the  alternate  terras. 
This  limit  is  numerically  superior,  but  not  algebraically  or 
in  fact. 

INFERIOR   LIMIT   OF   NEGATIVE    ROOTS. 

Art.   62.     Take  the  reciprocal  of  the  last  transformed 

equation,  that  is,  put  x  =  —  -,  and  find  a  superior  limit  of 

the  positive  roots  of  this  equation,   it  will  be  the  required 

limit,     because,     since   x  = ,  we  have  y  = ,  and 

y  X 

the  greatest  positive  value  of  y  will  correspond  to  the  least 

{numericaUy  considered)  negative  value  of  x . 

Newton's  limit. 

Art.  63 .  Any  number,  which  on  being  substituted  for 
the  unknown  quantity  in  the  first  member  of  an  equation 
and  in  its  derived  polynomials,  makes  them  all  positive,  is 
a  superior  limit  of  the  positive  roots. 

If  the  roots  a,  b,  c, I  of  f{x)  =  0  be  diminished  by 

x',  that  is,  if  we  make  x  =  x'-{-y,  we  shall  have,  eq.  (2), 
Art.  45: 


LIMITS  AND  PLACES  OF  ROOTS.  57 


=  0. 

If  such  a  value  be  i^laced  in  this  equation  for  x'  as  to 
make  all  the  terms  positive,  we  know  that  all  its  roots, 
that  is,  values  of  ?/,  must  be  negative,  and  from  the  rela- 
tion X  =  x'^ii,  we  have  y  =  x—.r\  so  that  y  being  nega- 
tive, x'^x,  and  consequently,  whatever  value  will  make 
the  first  derived  polynomial, /'(.r'),  positive  will  make  posi- 
tive the  original  first  member,  where  the  coefficients  are  the 
same,  but  x  takes  the  place  of  a  quantity  greater  by.'z;'. 

EXAMPLE. 

Find  a  sui)erior  limit  to  the  positive  roots  of  x^ — 5.r'^-[- 
lx—1  =  0. 

We  need  not  retain  the  dashes  upon^the  x,  but  write : 

f{oc)  =  dx^—Wx-\-l. 

1.2.3 

Beginning  at  the  last  derived  function  in  which  x  ap- 
pears, and  substituting  the  smallest  whole  number  which 
will  make  it  positive,  Ave  see  that  3  makes  it  positive.  Like- 
wise the  next  before  it,  and  so  on  to  the  last.  3,  then,  is 
the  limit. 

EXAMPLE   2. 

What  is  the  superior  limit  of  the  roots  of  x^ — 5.^* — 13a;' 
+ 17a;''— 69  =  0? 

We  have  derived  polynomicda  as  follow  (after  dividing  out 
their  appropriate  denominators): 
5a;^— 20./— 39a;'^4-  34r . 
IOj-^— 30./^— 39j'fl7. 
10a;'— 20a;— 13. 
5a'— 4. 
1. 


58  PRINCIPLES    OF    ALGEBRA. 

1  placed  for  x  gives  5 — 4=1,  positive;  but  fails  in  the 
next  above.  2  fails,  but  3  gives  a  positive  result.  3, 
when  tried  in/"(.r),  fails,  and  so  does  4,  by  a  single  unit, 
5,  being  tried,  gives  -[-,  and  being  tried  in  f\-r),  fails,  and 
so  does  6.     And  7  is  found  to  be  the  required  limit. 

It  will  be  perceived  that  this  has  given  us  the  smallest 
limit  in  whole  numbers,  and  it  will  always  give  us  a  closer 
limit  than  any  of  the  j)revious  methods.  The  amount  of 
comjDutation  confines  its  use  to  cases  where  closeness  of 
limit  is  im^^ortant.  It  was  invented  by  the  immortal  New- 
ton, who  has  shed  brilliant  and  enduring  light  upon  all 
of  the  man}^  branches  of  learning  to  which  he  addressed 
himself. 

Sudan's  test  of  imaginary  roots. 

Art.  64.  If  the  roots  of  an  equation  be  reduced  by  a 
quantity  r,  and  the  transformed  equation  shows  a  loss  of 
m  variations  of  signs,  and  if  the  reciprocal  equation  be  re- 
duced by  -,  and  this  transformed  equation  shows  n  varia- 
tions, which  were  not  lost  but  w^hich  remain;  then  there  are 
m — n  imaginary  roots  between  r  and  0. 

Because ,  in  reducing  the  roots  of  the  equation  by  /',  all 
positive  roots  less  than  r  will  have  become  negative*,  and 
there  will  be  as  many  positive  roots  between  0  and  r  as 
there  have  been  j)ositive  roots  changed  into  negative, which 
is  to  say,  as  many  as  there  have  been  variations  lost, 
whereas,  in  reducing  the  roots  of  the  reciprocal  equation 

by  -,  no  positive  root  greater  than  -  will  be  changed.   But 

should  a  different  result  appear,  it  would  indicate  the  ex- 
istence of  imaginary  roots,  the  number  of  which  within 
these  limits  will  be  the  number  of   variations  lost  by  the 

*  The  factors  of  the  first  degree  belonging  to  negative  roots  are  of  the  form  x-^c, 
x+d,  etc.,  and  in  the  multiplication  which  builds  up  the  first  member  of  the  reduced 
equation,  they  exercise  no  influence  on  the  signs  or  number  of  variations. 


LIMITS  AND  PLACES  OF  ROOTS. 


59 


first  transformation  minus  the  number  not  lost  in  the  one 
last  described. 

Now  suppose  /•  was  a  superior  limit  of  the  positive  roots; 
when  we  reduced  by  r,  the  number  of  lost  variations 
would  be  equal  to  the  number  of   positive  roots,  provided 

they  were  all  real.     And  in  the  recii)rocal  equation     would 

be  an  inferior  limit  of  the  positive  roots,  and  when  it  was 

transformed  by  reducing  the  roots  by  the   quantity    ,    the 

transformed  equation  would  show  no  loss  of  variations, 
provided  the  j^ositive  roots  were  all  real.  A  different  result 
would  show  that  there  was  an  absurdity  or  contradiction 
about  some  of  the  roots,  which  we  would  therefore  per- 
ceive to  be  imaginary.  And  these  would  appear  to  be  pos- 
itive. 

And  the  number  of  imaginary  roots  thus  discovered 
would  be  the  number  of  variations  lost  in  the  transforma- 
tion of  the  original  equation  minus  tke  number  not  lost  or 
which  remain  in  the  transf  or  oration  from  the  reciprocal 
equation. 

Again  take  the  original  equation  and  change  the  alter- 
nate signs;  the  positive  roots  will  be  turned  into  negative, 
and  the  negative  into  positive  roots.  Proceed  with  this  as 
with  the  original  equation,  and  we  shall  discover  the  num- 
ber of  imaginary  roots,  apparentl}^  negative. 

EXAMPLE . 

Find  the  number  of  imaginary  roots  in 

Since  this  equation  is  of  an  odd  degree,  with  the  abso- 
lute term  negative,  there  is  at  least  one  real  root  i)ositive, 
and  since  there  is  but  one  variation,  there  is  but  one  such 
root.  We  need  not  look  among  the  positive  roots  for  im- 
aginary roots,  but  according  to  Sudan's  Test,  we  change 


60 


PRINCIPLES  OF  ALGEBRA. 


the  alternate  signs  and  have 
which  1  is  a  superior  limit. 

COEFFICIENTS   OF    DIRECT 
EQUATION. 

1  —3  4-2  4-3—2  -h2  I  4-1 


.3j.^_|_2.r^  1  3,,^_2.r 4-2,  of 


COEFFICIENTS    OF    RECIPROCAL 
EQUATION  . 

2 --2  -}-3  -1-2—3  -hi  I  -f-1 


+1  —2  ±0  +3  +1 

+  2  ±0  +3  -1-5  +2 

—2  ±0  +3  +l,+3 
+  1  -1  -1  +2 

±0  -1-3  -rS  +2,-1-3 
+  2-^2+5  +10 

-1  -1  +2,+3 
_|_1  ±0  —1 

-1-2  +5-i-10,  +  12 

+2+4-^9 

±0-l,+l 

+1+1 

+  4+9,+19 

+  2+G 

+i,±o 

+1 

+6,+15 
+2 

+2 

+8 

1  +2  ±0  -1-1  -}-3  -f3 


2  -f  8-hl5+19-l-12-f  3 


Since  in  the  coefficients  of  the  transformation  from  the 
direct,  the  third,  ±0,  is  between  two  terms  of  like  signs, 
we  know  from  De  Giia's  Test,  that  there  are  two  imagin- 
ary^ roots  in  the  trAnsformed  equation;  we  may  therefore 
use  the  plus  sign,  which  shows  4  variations  lost.  In  the 
transformation  of  the  recij)rocal  equation  there  are  none 
left;   hence  4—0  =  4,  the  number  of  imaginary  roots. 

2.  In  QC" — 10jr*-[-6j:-[-l=:0,  how  many  imaginary  roots? 

Ans.     All  real. 

3.  How  many  imaginary  roots  in  .r* — 4..>;^-|-8.r^ — 16,r-[- 
20  =  0?  Ans.     None. 

4.  In  .r*  -f-.r'-f  j""^4-3.r — 100  =  0,  how  many  imaginary 
roots  ? 

Ans.     2  imaginary  roots  and  2  real  with  opposite  signs. 


PLACES    OF    REAL    ROOTS. 


Art.    65.     It  has  been  shown  that  if  x  varies  under  the 
law  of  continuity  that /(.:tT)=a;'"-[-P.r"'-'-f  .  .-J-T^-j-  f/will  do 


LIMITS  AND  PLACES  OF  ROOTS.  61 

SO  likewise.  Let  us  sii])pose  that  we  had  substituted  for  x 
in  /(.*•)  a  number,  p,  and  the  result  was  greater  than  0,  or 
-{-.  Then,  if  x  decreases  under  continuity,  it  will  after  a 
time,  come  upon  the  value  of  one  of  the  roots,  when  the 
result  will  be  0.  Continuing  to  decrease,  its  value  (say  7) 
will  give  a  result  less  than  0  or  —  and  consequently  we 
say  that  if  two  number's,  p  and  q,  when  miihd'diited  for  the 
unknown  quanUty  in  the  first  member  of  an  equation  of  which 
the  second  member  is  0,  give  results  with  opposite  signs,  there 
is  at  least  one  real  root  tjetween  p  and  q. 

A  quantity  may  change  its  sign  by  passing  through  in- 
finity as  well  as  through  0.  Let  :r  =  -;  here  as  y  decreas- 
es, .r  increases;  when  //  is  very  small,  r  becomes  very 
great;  when  //  =^  0,  .v  =z  :c  ;  when  ?/<0,  or  negative,  x  be- 
comes negative;  but  in  the  rational  integral  function, 
which  is  the  first  member  of  the  equation,  no  finite  value 
of  X,  as  between  p  and  7,  could  make  f{x)  =  cc  . 

There  might  be  more  than  one  root  between  p  and  q. 
Moreover,  if  there  are  roots  between  p  and  q,  the  substitu- 
tion of  p  and  q  will  not  necessarily  produce  results  with 
contrary  signs,  for, 

Art.  66.  When  an  odd  number  of  routs  lie  between  p 
and  q,  their  substitution  loilt  give  results  having  opposite 
signs;  when  an  even  number  of  roots  lie  between  them  the 
results  will  have  tJie  same  signs. 

Suppose  that  there  were  several  roots,  a,  b,  c,  etc.,  be- 
tween 2^  aiicl  q,  and  some  others  besides.  Let  the  product 
of  the  factors  of  the  first  degree  with  respect  to  these  latter 
roots  be  Y;   then  we  shall  have : 

/(,r)  =  {x-a)(x-b){x-c) X  r=  0. 

Substitute  for  x  first  p  and  then  q;  let  Y'  be  what  Y  be- 
comes on  the  substitution  of  p,  F"  the  result  of  substitut- 
ing q  in  Y.  Now  Y'  and  F"  will  have  the  same  sign; 
otherwise,   by  Art.   65,  there  would   be   another  root,  or 


62 


PRINCIPLES  OF  ALGEBRA. 


roots,  lying  between  p  and  7,  which  is  contrary  to  the  sup- 
position. 

If  we  now  make  the  sul)stitiitions,  and  for  convenience 
write  the  tirst  result  over  the  other,  we  shall  have: 

{p~a){p—mp—^^) X>" 

{ri—a)l<i-h)(q—c) X  y' 

or  otherwise  thus: 

p — a      p — h      p — c  F' 

q—a  ^  q—b  ^  q—i-  '^  '  \ ^^    F'' 

Suppose  7>>7,  then  a,  b,  c,  etc.,  willjbe  <Cj>  and  >7,  so 
that  all  the  quotients  will  be  negative  except  ,7,,. 

Now,  if  the  number  of  roots  a,  b,  c,  etc.,  between  p  and 
q  is  even,  the  product  of  these  fractions  will  be  positive, 
and  the  first  result  divided  by  the  second  will  have  a  posi- 
tive quotient,  that  is,  the  results  of  the  substitutions  of  p 
and  q  will  have  the  same  sign,  and  the  contrary  will  be 
true  when  ct,  b,  c,  etc.,  are  odd  in  number. 


THE    THEOREM    OF    STURM. 

Art.  67.  But  the  best  of  all  the  modes  3'et  discovered 
of  determining  the  character  and  places  of  the  roots  of  an 
equation  is  the  celebrated  theorem  of  Sturm,  contributed 
in  1829  to  the  scientific  world  by  that  eminent  French 
mathematician.  The  object  of  *S/wrm\s  ilieorein  is  to  dis- 
cover the  number  of  real  and  imaginary  roots  in  any  eqwx- 
tion,  and  the  places  of  the  real  roots. 

Sturm's  Theorem  does  all  that  is  accomplished  b}^  the 
methods  which  have  thus  far  been  examined,  and  more  be- 
side. Still  those  methods  should  be  preserved,  because 
they  are  sometimes  sufficient  for  the  purpose  in  hand  and 
of  easier  application  than  the  theorem  of  Sturm . 

This  theorem  deals  with  the  signs  of  certain  functions  of 
the  unknown  quantity,  which  are :  the  first  member  of   the 


LIMITS    AND    PLACEfi    OF    ROOTS.  63 

equation,  its  first  derived  polynomial  and  certain  others 
which  are  formed  in  the  following  manner:  First  free  the 
equation  of  equal  roots,  if  it  has  any;  then  apply  to  /(.r), 
the  first  member,  and  to/'(^),  its  first  derived  j^olynomial, 
the  process  for  finding  the  H.C.D.,  but  with  this  differ- 
ence— after  each  remainder  has  been  found,  change  its  sigri, 
and  during  the  intermediate  operations  neither  introduce 
nor  suppress  any  factor  but  a  pcMtive  one. 

Let  /(.r)  =  Nx"^  -L.rx'"-'^Q.r"'-'-\- -[■Tx'\-.  U  =^  0  be 

the  equation,  and  designate  f{x)  by  V,  and/'(.r)  by  Vi,  and 
by — V^, — T3, — r^, — Tv,  the  remainders  of  the  various 
divisions  wherein  the  quotients  were:  (>,,  Q.,,  Q^, ....  Q^-j- 

We  shall  have  the  following  equations: 


F_=  F_a._-T; (1) 

Tv  cannot  be  0,  otherwise  there  would  be  a  CD.  between 
f{x)  and  /"(./•),  which  is  contrary  to  the  supposition .  It 
must,  then,  be  a  number,  because  the  oj^eration  is  to  be 
carried  on  until  the  last  remainder  is  independent  of  r. 

Art.  68.  Letyl  and  B  be  two  numbers,  and  A<^B.  Let 
A  be  substituted  for  x  in  the  expressions,  V,  F,,  K^,  T^, 
etc.,  and  the  signs  of  the  results  recorded;  then  substitute 
J9  and  record  the  signs.  The  sign  of  K,.  will  always*  re- 
main the  same,  being  independent  of  x.  Then  the  theorem 
declares  that 

The  number  0/  varlal  Ions  in  the  first  series  of  signs,  dimin- 
ished by  the  number  of  variations  in  the  second,  will  be  equal 
to  the  number  of  real  roots  between  A  and  B. 


64  PEIIS'CIPLES    OF    ALGEBEA. 

Art.    69.     To   show  this,  it  will  be  convenient  first  to 
establish  three  lemmas,  as  follow : 

FIRST. 

No  two  coxHi'vutire  functions,  V,  Fj,   F,  etc.,  ran  hccoinr  0 
for  the  same  value  attributed  to  x. 

Let  us  take  any  equation  out  of  the  group  (1),  as 

and  suppose  that  V„_i  and  F„  should  both  vanish  for  a 
value  of  x,  then  from  the  equation,  F;,.y.i,  would  also  be 
zero.  And  the  next  equation  of  the  series,  having  F,,  and 
Kt-i-u  both  0,  would  give  F„.|._,  =  0,  and  so  on.  Thus  they 
would  all  vanish  and  the  last  equation  would  give  F,.  =0, 
which  cannot  be . 

SECOND. 

Art.    70.      When  amj  one  of  these  functions  becomes  0, 
the  one  before  it  ivill  have  a  different  sign  from   the  one  fol- 
lowing it  for  the  same  value  of  x.  as  is  shown  by  taking  any 
one  of  the  equations,  as   F.^-^  Fs  Q^ —  K*  and  letting  F^  =0 
.  F  = V 

THIRD. 

Art.  71.  If  a  number  almost  equal  to  one  of  the  real 
roots  of  the  equation  be  substituted  for  the  unknown  quantity 
in  the  first  member,  and  likewise  in  its  first  derived  polynom- 
ial, the  results  will  have  contrary  signs;  but  if  the  substituted 
quantity  be  greater  than  this  root  by  an  extremely  small 
amount  the  signs  of  the  results  will  be  the  same. 

Let  us  suppose  that  a  were  a  root  and  the  added 
small  quantity  ii.  Let  a-\-u  and  a — u  be  substituted  for  .'■: 
then  b}^  (2)  Art.  45  we  shall  have: 

/■(«+")  ==/(«)+/'(")«+/"(«)  j^-f 


THEOREM  OF  STURM.  65 

f(a-u)  =  /•(«)  -r{a)u.+f"(a)^'^-. .  . . 

ytn—i 

In  these  f{a)  -=  0,  and  as  a  is  a  very  minute  quantity, 
the  first  terms  of  those  portions  of  the  series  which  remain 
will  far  exceed  in  value  the  sum  of  the  remaining  terms 
(Art.  54)  and  will  give  sign  to  the  series.  These  series  will 
then  become: 

«[/'('')+l"2  /"'(«)+.■  ■  •  •  +  «"-']  ■  •  •  -(3) 

and        -u\f'(ay-^^f"(a)+  .  . . ,  +  «'-'] ...  .(4) 

The  upper  one  will  be  positive  and  the  lower  will  be 
negative,  and  /'(«)  positive  all  the  time,  as  they  are  here 
shown.  In  all  cases,  it  is  apparent  that  the  sign  of  the 
lower  series  will  be  different  from  f\a),  and  that  of  the 
upper  the  same.  But  (4)  and  (2),  which  are  the  same,  are 
the  result  of  substituting  a — u  in  F,  while  (3)  and  (1)  are 
the  result  of  substituting  a-\-u.  f\a)  is  the  result  of  sub- 
stituting a  for  X  in  Fi.  The  truth  of  the  lemma  is,  then, 
demonstrated. 

Art.  72.  Then  to  demonstrate  the  theorem  of  Sturm, 
let  us  suppose  a  varying  quantity,  A^  which  at  the  outset 
is  less  than  the  least  of  the  real  roots  of  F  =  0,  Fi  =  0, 
F^  =-  0, .  . . .  F„  =  0, . . . .  and  F,_i  =  0,  that  is,  of  all  the 
equations  formed  by  putting  Sturm's  functions  equal  to  0. 
Let  it  be  substituted  for  x  in  all  of  them,  and  record  the 
signs  of  the  results.  Afterwards  suppose  A  to  grow:  after 
a  time  it  will  be  equal  to  the  least  of  the  roots  mentioned 
above,  and  some  one  of  the  functions  Fi,  Tg,  etc.,  will  van- 
ish. But,  as  its  sign  agrees  with  the  one  before  it  and  dis- 
agrees with  the  one  after  it  (Art.  70),  or  else  agrees  with 
the  one  behind  it  and  disagrees  with  the  one  before  it,  the 
number  of  variations  will  not  be  affected. 

And  this  will  be  true  even  if  two,  or  several,  of  the  func- 


66  PRINCIPLES  OF  ALGEBRA. 

tions  vanisli  at  the  same  time;  because  the  same  conditions 
as  those  just  described  would  hold  when  the  vanishing  func- 
tions were  separated  from  each  other  by  intervals  in  the 
series  of  them.  And  this  must  be  the  case  because  no  two 
consecutive  ones  can  vanish  simultaneously  (Art.  69).  This 
will  continue  until  the  varying  value  of  A  arrives  in  the 
close  neighborhood  of  a  root  of  V  =.  0,  that  is,  the  origi- 
nal equation,  when  the  signs  of  V  and  Fj  will  be  different, 
giving  a  variation,  and  after  it  passes  the  value  of  the 
root,  and  — u  becomes  -{-u,  they  will  agree  in  sign,  giving  a 
permanence,  or  losing  one  variation  (Art.  71). 

If  it  be  supposed  that  A  continues  to  grow  until  it  ar- 
rives in  close  proximity  to  another  real  root  of  T=  0,  the 
same  thing  will  take  place,  and  when  V  becomes  zero  and 
emerges  with  a  change  of  sign,  it  will  have  passed  another 
rooty  ayid  another  variation  will  have  been  lost . 

And  so  every  time  a  real  root  is  passed,  a  variation  of 
the  signs  of  the  functions  F,  Fj,  Fg,  etc.,  will  have  been 
lost,  until  A  has  grown  greater  than  the  greatest  real  root 
of  the  original  equation.  The  number  of  variations  lost 
will  be  equal  to  the  number  of  real  roots  between^  and  B, 
and  as  they  were  taken  as  the  numerically  superior  limit  of 
the  negative  roots  and  the  superior  limit  of  the  positive 
roots,  the  total  number  of  real  roots  will  be  known.  This 
number  subtracted  from  the  exponent  which  shows  the  de- 
gree of  the  equation  will  give  the  number  of  imaginary  roots. 
This  last  must  of  course  be  0  or  an  even  number,  and  thus 
the  theorem  is  found  to  be  true . 

Art.  73.  If  in  finding  Fj,  F^,  etc.,  any  one  of  the 
functions  placed  =  0  should  give  an  equation  all  of  the 
roots  of  which  were  imaginary  (and  this  fact  would  be 
known  by  the  function  remaining  of  one  sign  for  all  real 
values  of  x,'^  the  work  need  proceed  no  farther. 

The  polynomial  function  remaining  always  remaining  of 

*  Note.— When  the  first  member  is  constantly  of  the  same  sign  for  all"  real  values 
of  X,  we  infer  that  all  the  roots  are  imaginary,  because  if  one  value  resulted  in  plug 
and  another  in  minus,  there  would  be  a  real  root  between  the  numbers  substituted. 


THEOBEM  OF  STURM.  67 

the  same  sign  and  the  last  one,  F^,  being  constant,  none  of 
those  between  them  can  change  sign.  And  therefore,  if 
any  loss  of  variations  takes  place,  it  must- do  so  among 
those  which  precede.  This  conclusion  will  be  reached  by 
considering  the  chain  of  equations  which  connect  the  poly- 
nomial spoken  of  with  V^. 

Both  ends  of  the  chain  remaining  of  a  constant  sign,  the 
nature  of  the  connection  is  such  as  to  prevent  all  interme- 
diate functions  from  changing.  Or  otherwise  thus:  Take 
the  equations 

'r—4:  ==  'r—3  Vr-5  '  r—2  •  •  •  •  (1) 
'r-3  ^^  ''^r-2  Vr-2  '  /--/  ....  (2) 
Vr..-^     =         Yr.-.       Qr...     -     V, (3) 

Suppose  that  it  was  found  that  F^..2  would  not  change 
sign.  Since  the  quotients  ^i,  Q.^  ....  Q^..j  would  have  no 
influence  on  the  sign,  as  the  values  of  x  are  not  substituted 
in  them,  if  we  transpose  V^  to  the  first  member  then  thai 
member  being  fixed  in  sign,  F^..i ,  would  be  so  likewise. 

If  F^..3  were  found  constant  in  sign  we  might  eliminate 
V^i  out  of  (2)  by  substituting  its  value  (3)  and  a  single 
equation  would  result  with  F^_2  and  quantities  fixed  in 
sign;  hence  F^_2  could  not  change  sign  and  in  the  same 
way  it  could  be  shown  that  F^_i  could  not  change  sign. 

If  V^_i  were  found  to  be  constant  in  sign,  F^_3  and  V^_^ 
could  be  eliminated  by  substituting  the  values  from  (2)  and 
(3)  whence  it  would  be  seen  that  F^_i  must  not  change 
sign;  then  in  succession  the  other  two.  And  so  for  any 
function. 

Art.  74.  After  the  varying  value  of  x  has  passed  above 
and  below  the  greatest  and  least  roots,  the  further  increase 
or  decrease  can  make  no  difference  in  the  signs;  and  this  is 
true  up  to  any  extent  even  to  -f  oo  and  —  oo .  The  sub- 
stitution of  -j-  X  and  —  x  v^ill  cause  the  functions  to  take 
their  signs  from  their  first  terms,  and  will  be  found  conve- 
nient because  we  need  substitute  in  the  first  term  only . 

This  will  give  the  number  of  the  real  roots .     But  in  ad- 


68 


PRINCIPLES    OF    ALGEBRA. 


dition  to  this  we  may  wish  to  know  their  places,  that  is,  be- 
tween what  whole  numbers  they  may  lie .  To  do  this  we 
substitute  for  J  and  B,  0  and  1,  1  and  2,  2  and  3,  3  and  4, 
etc.  Thus  having  found  one  or  more  roots  between  2  and 
3  we  know  that  the  roots  are  2  -\-  sl  fraction  less  than  unity. 
For  the  negative  roots  we  substitute  —  1  and  0,  —  2  and 
—  1,  —  3  and  —  2,  and  so  on.  Having  found  a  root  be- 
tween —  3  and  — 2  it  will  be  ^ — ^3  plus  a  fraction,  etc.  Its 
initial  figure  will  be  — 2,  as  — 2.57. 


EXAMPLE. 

1. 8^-3  _6j;— 1=0.    f'{x)=8a^—6x—l 

f^(x):=24:x'^ — 6.       Suppress  the  positive 
factor  6  in/''(^)  and  we  have  Fi=4^"^ — 1. 
8a^—6x—l  I  4:x'—i 
8a?~-2x  I  2x 

— 4,x — 1.       Here    changing    sign    we    have 
4:X-\-l=V2.     Multiplying  V^  by  the  positive  factor  4  and 
we  get  16x^ — 4  to  be  divided  by  4.:r-f  1. 
16^^—4     I  4a:+l 
16ar^+4ar      |  4a:— 1 

4:X — 4 

—40^—1 

— 3  and  -\-  3  and  -f  3  =  F"^.     Hence  the   poly- 
nomials are 

minus  oo  .  plus  go  . 

V==Sx'—6x—l     — 
F,=  4a^— 1  -f 

V,=  4.x  -f  1  — 

Vr=       ■      +3  + 

Hence  3  variations  have  been  lost  and  all  the  roots  are 
real.  To  determine  their  places:  substitute  0  and  1,  0 
and  —  1 . 


3  variations. 


0  variation 


0  and  1 

— 

+ 

~ 

-f- 

+ 

-f 

+ 

+ 

One  variation  lost  and  hence  there  lies 
between  1  and  0  one  real  root,  which  is 
zero  plus  a  fraction.  There  are  no  more 
positive  real  roots  becjiuse  -f-  1  gives  the 


THEOREM  OF  STURM. 


69 


same  signs  as  -f- 
tween  1  and  -{-  oo 
0  and  —  1 


00  ,  and,  therefore,  no  real  root  can  lie  be- 


+ 


Two  variations  lost,  and  there  are  two 
negative  roots,  between  —  and  0.     We 
have,  then,  3  roots,  1  positive  and  2  neg- 
_j_  -|-  ative.   The  signs  being  the  same  for  —  1 

and  —  CO  we  know  that  plus  1  and  min- 
us 1  are  the  smallest  limits  in  whole  numbers. 
Example  2.     x'  —  4.x^  —  6^  -f  8  =  0. 

Here  V  =  x^  —  Aaf  —  Gx  -f  8 
F,=  dx'  —  8^  —  6 
Multiplying  F  by  the  positive  factor  3,  and  proceeding 
as  above  indicated. 
dar^—Ux'— 18x^24:  I  3j?»— Sot— 6 


3^^- 

Sx'—Qx.               x,—l 

4:X^ — 12er-}-24;     suppress  -|-4  and  multiply  by  3: 
3^^  8^+  6 

—17^+12     .-.    V,=  llx—12 

S,jc'—Sx—6  1  17.7r— 12 
17                    3^,-100 

Slar'^— 136^—102 
Blx'—  S6x 

— lOO^r— 102 
17 

—1700^—1734 
—170007—1200 

—  534 
Then  we  have: 

V  =  af—lx'—Gxi-S 

Fi  =  3a;'^— 8^— 6 
V,  =  llx—12 
V,  =  +534. 

For  X  =  -{-oo  ,+  +  +  +,  no  variations. 

For  X  =  — 00  , 1 1- ,  3  variations. 

3 — 0  =  3  variations  lost,  and  3  real  roots. 


70 


PRINCIPLES  OF  ALGEBRA. 


V    V,    K    Fa 

Var. 

For  x  =  0,  -\ \- 

2 

x  =  l, +   -f 

1 

x=2, +  + 

1. 

x  =  3, +  + 

1 

^  =  4,  -  +  +  + 

1 

^=5,  +  +  +  H- 

0 

^  =  0,  + + 

2 

x=—l,-\-  + h 

2 

x=—2, \ h 

3 

Between  — 2,  which  gave  the  same  signs  as  — x  ,  and 
— 1,  there  was  1  variation  lost;  hence  the  root  is  — 2  -[-  a 
fraction,  or  a  root  whose  initial  figure  is  — 1.  Between  0 
and  -f  1  a  variation  was  lost,  and  there  is  a  root  whose 
initial  figure  is  0.  At  4  there  was  1  variation  and  at  5 
none;    hence  a  root  4  -f-  a  fraction. 

Ex,  3.     2x'  —  11./;=  -f  8x  —  16  =  0.     Here 
V=  2^*  — ll^^-f  8^  —  16 
V,=  4:x:'  —  11^  +4 

V,^^nx'—12x  +32.  If  this  were  placed 
equal  to  zero  the  two  roots  would  be  found  to  be  imagin- 
ary. If  the  trinomial  were  a  true  square  we  should  have 
4(11^^  _|-  32)  =  ( —  12xy  which  it  is  not.  V,  will  not 
change  sign  for  any  real  value  of  x  and  we  will  proceed  no 
further  in  getting  Sturm's  functions . 
-f-  ^  gives  -j-  -\-  -]-;  no  variation 

—  00  gives  -j- [- ;  2  variations   .  •  .   2  —  0  =  2  and 

there  are  2  real  roots,  and  of  course  2  imaginary  roots. 

X  ^=  0  gives ^   -j-         X  =       0  gives \-  -{- 

X  ==  1  gives \-         X  =  —  1  gives  —  +  + 

X  =  2  gives  —  +  -h         ^'  =  —  ^  gives [- 

^  =  3  gives  +  +  +         oc  =^  —  3  gives  -|- 1- 

and  the  initial  figures  of  the  real  roots  are  2  and  —  2. 

Ex.  4.  x^  -j-  11^^  —  102^  +  181  =  0;  in  which  Uvo  of 
the  roots  are  nearly  equal. 


THEOREM  OF  STURM.  71 

The  functions  are   V  =  x^  +  llaf  —  102x  -f  181 

V,  =  Saf -\-122x  —  102 

V,  =  12207  —  393 

F^=  -\-  number. 
—  00  gives  3  variations  and  -f-  oo  none;  so  there  are  3  real 

roots.     X  =  0  gives  -j -f-  and  so  do  a?  =  1,  2  and  3, 

but  X  =  4:  gives  no  variation,  therefore  there  are  two  posi- 
tive roots  lying  between  3  and  4.    Their  initial  figures  are  3. 

Let  the  equation  be  transformed  into  one  whose  roots  are 
less  by  3  (Art.  45).     The  functions  of  this  equation  will  be: 

V,  =  3if^4:0y—9 
V,  =  122?/— 27 
Fa  =  -|-  number. 

Now  in  these  substitute  y  =  0,  y  =  0.1,  y  =  0.2,  y  =  0.3, 
etc.,  and  we  find: 

y=   0   gives    +    —   —    -f-,     2  variations; 

y  =  .1   gives   -f    —  —   -f ,     2  variations; 
•    y  =  .2   gives    -|- -f,     2  variations; 

2/  =  .3   gives    -}-    -j-    -j-    -|-,     no  variation; 
two  positive  roots  between  .2  and  .3,  and  of   the  proposed 
equation  between  3.2  and  3.3. 

Transform  the  equation  into  another  whose  roots  shall 
be  less  than  the  roots  of  the  last  by  0.2,  and  we  have: 

F  =  8^-|-(20.6).s'^— (•88)8+.008 
F,  =  'ds'-\-(4:1.2)s—.88 
F,  =  122s— 2.6,  or  61s— 1.3 
Fz  =  -\-  number. 
Substitute  s  =  0,      the  signs  will  be : 
s  =  .01,  the  signs  will  be: 
s  =  .02,  the  signs  will  be : 
s  =.  .03,  the  signs  will  be 
One  positive  root  between  .01  and   .02,  also  one  bet\wen 
.02  and  .03,  and  for  x  we  have  a;  =  3.21  and  x  =  3.22. 
Their   sum  =  6 .  43.        .  • .    _  H  _  6.43  =  —17 .  43,  =  the 
third  root,  which  is  negative . 


+  —  +, 

2  var. 

+  —  +, 

2  var. 

h> 

1  var. 

+  +  +  +, 

no  var. 

72  PRINCIPLES   OF   ALGEBRA.. 

CHAPTER  YI. 

EXACT    AND    DIRECT    SOLUTION    OF    EQUATIONS! 

Art.  84.  We  know  that  equations  of  tlie  first  and  sec- 
and  degrees  admit  of  direct  and  exact  solutions;  the  first 
presenting  a  single  root  and  the  latter  two  roots . 

It  will  be  now  shown  that  such  equations  of  the  third  de- 
gree as  have  two  imaginary  roots  can  be  solved  direc  Jy  and 
exactly;  and  that  equations  of  the  dth  degree  of  ivhich  two, 
and  only  two,  of  the  roots  are  imaginary  can  be  directly  and 
exactly  solved.  Above  these  equations  there  are  no  means 
of  exact  and  direct  solution,  at  least  none  as  yet  have  been 
discovered,  and  it  is  believed  that  none  can  be  discovered . 
Propositions  for  the  exact  solution  of  equations  of  degrees 
higher  than  the  4th  have  occasionally  been  presented,  with 
plausibility,  but  the  practical  results  have  been  such  as  not 
to  invalidate  the  accuracy  of  the  statement  above. 

Art.  85.  Let  x^  -{-  Px''  ^  Qx  -\-  M  =  0;  this  is  a  gen- 
eral representative  of  equations  of  the  3d  degree,  and  any 
equation  of  the  3d  degree  will  be  a  particular  case  of  this 
general  form. 

But  as  the  difficulties  of  solving  equations  of  a  degree 
higher  than  the  second  are  sufficiently  great  at  best  it  will 
be  well  to  diminish  them  by  removing  as  many  terms  as 
possible. 

Now  as  we  can  exchange  any  complete  equation  for  an- 
other wanting  the  second  term,  let  us  write 

•  x^  -\-px  -}-  q  =  0 (1) 

and  form  for  it  the  functions  of  Sturm,  we  shall  have 
x^-^-jJX-^q  I  3x^-{-p 
3  ~    U 


V=3(^-\~px-\-q 
Fi^3^^+^ 
V^=  —2px—Sq 
V,=-4:p'—21q' 


Sx'-\-dpx^Sq 

3x^-\-px 


2px^dq .  • .  F,^—2px—Sq 


cardan's  soldtion.  '      73 


Asrain : 


3./^+p                  1  -2 
2p                             1- 

6px'-\-dqx 

oxSq 
-3-r,  9^ 

—dqxi-2p' 
2p 

—18pqx-\-4:p^ 
—18pqx—27q' 

4:2f^21q 

-27(/ 

The  roots  of  this  equation,  it  being  of  the  3d  degree,  must 
all  be  real,  or  else  one  must  be  real  and  two  imaginary. 
Since  2^  and  q  are  not  numbers,  but  the  general  representa- 
tives of  any  real  coefficients  of  the  first  and  zero  powers,  let 
us  see  what  relations  they  must  have  in  order  that  all  the 
rooU:  shall  be  real.  There  must  result  from  the  substitution 
of  —  00  3  variations,  and  from  the  substitution  of  -|-  00  no 
variations,  in  order  that  all  the  roots  may  be  real.  When 
-f  00  is  substituucd  in  V  the  result  is  -f ;  in  V^  it  is  -f ,  but 
in  V2  it  will  not  be  -[-  unless  p  is  negative.  Then  p  must 
be  negative;  but  ^  will  not  be  -\-  unless p  is  negative  and 
moreover  has  such  a  value  that  4j/  is  greater  than  27r/. 
With  this  condition  F3  will  be  plus  and  remain  so.  The 
first  term  of  K  will  also  have  a  positive  coefficient;  so  that 

the  substitution  of  —        will  give  the  signs 1 \-;  3 

variations,  and  the  substitution  of  -|-  ^  gives  no  variations. 

We  see,  then,  when  ^-  >  |  or  v-j  >  ft)  that  all  the  roots 
will  be  real. 

cardan's  solution  of  equations  of  the  third  degree. 

Art.  86.  If  the  equation  is  complete,  let  it  be  trans- 
formed to  another  wanting  the  second  term .  It  will  take 
the  form 

a^^px-i^q  =  0, (1) 

10 


74  PRINCIPLES  OF  ALGEBRA. 

Let  the  unknown  quantity  be  placed  equal  to  the  sum  of 
two  other  unknown  quantities,  ij  and  z;  then  x  =  y-\-z; 
x'  =  ]f^2yz{y^z)^z'  .-.  ;f^-dy{,j^z)2-{y'^z')  ^  {),  iin&* 
replacing  y-\-z  by^^  in  the  second  term,  we  have: 

^_3^,^-(y;3j„,3)_0, (2) 

This  has  the  same  form  as  eq.  (1),  and  by  comparison: 

p  =  -Syz, ... (3),  and  q  =  -(y'+^),  or  y'-^^  ^-q,.. (4) 

— P  — P^ 

Since  2  =  .,    ,  2^  =^  ^rr\,   and  this  in  the  value  of   — q 
Sy  21i/'  ^ 

gives  y— ^3  =  —q;  •  •  •  y'-{^qy'  =  ^■ 

Since  this  is  a  trinomial  equation,  we  have: 


'^-'V-lMf+D 


and,  since  the  equation  y-^z  =  ^  is  symmetrical  with  re- 
spect to  y  and  z,  z  will  have  the  same  values.  But  not  to 
repeat  the  same  value  for  both,  we  will  take  the  first  for  y 
and  the  second  for  z,  and  adding  them  together,  we  get  for 
the  value  of  x: 


which  is  the  celebrated  Formula  of  Cardan. 


(f  ^  If 

4  "^27, 


Under  each  grand  radical  sign  there  is  the  same  indi- 
cated square  root,  which  will  be  real  when  p  is  positive, 

p3 
and  also  when  p  is  negative,  provided  that  ^  is  less  than 

J-.     2"  is  of  course  always  positive.     But  the  inequation 

~  <^  J-  >  ^y  -^I't-  '^^j  shows  that   all  the  roots  cannot  be 

real,  therefore  one  loill  be  r^eal  and  two  imaginary .   Cardan's 
formula  will  then  solve  the  cubic  equation  in  such  a  case. 
If  the  last  inequation  should  prove  to  be  an  equation  in 


cardan's  solution.  75 

any  case,  then  the  indicated  square  root  would  vanish  and 
X  =  2UJ — I ;   which  is  real,  and  Cardan's  formula  would 

apply  in  this  case  also.  But  if  we  seek  the  greatest  com- 
mon divisor  between  the  first  member  of  the  equation  and 
its  first  derived  polynomial,  the  remainder  is  4:p-{-21(f,  and 
if  this  =  0,  there  is  a  CD.  of  the  first  degree  with  respect 
to  .r,  and  therefore  two  of  the  roois  are  equal. 

If  all  three  of  tlie  roots  were  equal,  the  equation  would  re- 
duce to  a  binomial  equation  of  the  form  (.r — of  =  0,  and 
X  =  a,  and  all  the  roots  would  be  =  a. 

But  when  p  is  negative  and  jy^^j^,  Cardan's  Formula 

fails,  and  this  is  the  condition,  that  all  the  roots  are  real,  but 
not  equal.     All  are  unequal. 

Art.  87.  Since  every  quantity  has  3  cube  roots,  and 
since  x  =  sum  of  two  cube  roots,  it  might  at  first  sight  ap- 
pear that  the  cubic  equation  had  nine  roots,  because  each 
one  of  the  first  set  might  be  taken  in  conjunction  with  each 
of  the  3  in  the  second  set,  making  9  in  all.  Now  we  know 
that  every  equation  of  the  third  degree  has  three  roots  and 
no  more,  and  this  appearance  of  niae  roots  must  be  de- 
ceptive. 

To  explain  this,  let  us  remember  that  the  three  cube  roots 
of  any  quantity,  as   a^   are  of  the  form  a,   — ^ and 

~ '- ',  and  so  of   the  values  of  y  and  z.     But  it  will 

be   remembered   (eq.  3,  Art.   79),    that  ^2  =  — ^,    a  real 

o 

quantity,  since  p  and  q,  the  coefficients   of   the   proposed 

equation,  are  supposed  to  be  real.     If,  then,  we   take  one 

of  the  cube  roots  which  are  equal  to  y  and  one  of  the  cube 

roots  which  are  equal  to  z  to  make  a  value  of   x,  that  is,  to 

make  up  a  root  of  the  equation,  we  must  so  take  them  that 

their  product  will  be  real.     We  could  take  the  two  rational 


76  PKINCIPLES    or    ALGEBRA. 

and  real  cube  roots,  but  when  we  take  an}'  one  of  the  imag- 
inary expressions  we  must  take  another,  but  differing  in 
sign  before  the  radical  part  of  it,  so  that  the  product  would 
be  the  difference  of  two  squares  and  real.  This  restriction 
would  allow  us  to  form  two  values  only  of  x  out  of  the  im- 
aginary parts,  which  with  the  one  made  up  of  the  real 
parts  would  give  the  number  of  values  of  x,  three,  and 
only  three. 

In  the  solution  of  numerical  examples,  it  will  in  general 
suffice  to  substitute  in  Cardan's  Formula  for  q  and  i^  their 
appropriate  values;  but  sometimes  greater  simplicity  may 
be  obtained  by  treating  the  example  in  the  same  manner 
that  Qi?^'px^q  =r  0  was  treated  in  the  deduction  of  Car- 
dan's Formula. 

EXAMPLES. 

1.     Solve  the  equation  a^-  9j^-f  28./— 30  -::  0. 

Ans.    3,  3  f  1— 1,  3— V  ^. 

The  transformed  equation  is  i/^-f  ^  ="  O?  ^i^  which  p  --=  1 
and  5  =^  0    .  • .  Cardan's  Formula  becomes: 


— •^•\/l/(a7)     •'•    ^^  =  0..  one   root,  and  as  x  =  w-f  3,   x 

=  3.     This  "  divided  out"  of  the  equation  leaves  an  equa- 
tion of  the  second  degree  with  roots  as  above. 


2.     Solve  the  equation   ,r^— 7.r'-f  14r— 20  =-  0. 

Ans.    5,  1+1   — 3,  1 — I   — 3. 

,     7       344        ^ 

The  transformed  equation  is    //' — ~u — -^^   ==  0. 

And  this  cleared  of   fractions   gives:  ?/  —  21?/  —  344  =  0. 
One  root  of  this  is  8  and  w=  v       8    ^     ,  ,7       8-|-7 

=  5. 


waking's  method.  77 

Ex.  3.     Solve  x^  —  6.^^^  +  lO.r  —  8  =  0. 
Here  ic^  —  ^u  —  4  ==  0,  and  the  two  radicals  in  Cardan's 
formula  give  1.5774-  .  .  .   and  0.422-f ;  their  sum  is  2  =w. 
Ex.  4.     ^  +  1.r  +12  =  0. 
Ex.  5.     x^  —AHx^-'.  128. 
Ex.  6.     a''  —  3./^  —  2.V'  —  8  =  0.     Let  a.-^  =  ^. 

Remark — When  ^_<;^^  and  the  latter  is  negative,  the  values   of  x 

are  apjyareyitly  imaginary  though  known  in  fact  to  be  all  real.  This  is 
called  the  irreducible  case  because  Cardan's  formula  fails,  and  no 
means  have  yet  been  discovered  to  surmount  the  difficulty  by  Algebra 
alone. 

SOLUTION    OF    EQUATIONS     OF     THE     FOURTH     DEGREE. 

Art.  88.  The  equations  of  this  degree  admit  of  direct 
and  exact  solution  by  the  methods  now  known  only  when 
they  have  hvo,  and  only  two,  imaginary  roots.  If  their  roots 
are  all  real  or  are  all  imaginary  we  do  not  know  how  to 
solve  them  exactly,  but  in  the  case  of  numerical  equations 
can  resort  to  some  method  of  approximation. 

Descartes  and  Waring  have  each  demonstrated  an  excel- 
lent method  of  solving  equations  of  the  fourth  degree  hav- 
ing only  tw^o    imaginary   roots.     We  will   here  reproduce 

DR.  WARING's    METHOD    OF    EXACT    SOLUTION   OF    EQUATIONS    OF  THE 
4th    DEGREE. 

Art.    89.     Let  the  proposed  equation  be 

^*  -f  2pr^  =  qx^  -f  rx  -f-  s (1) 

Now  {x'  +  pr  -t-  ny  =  x*  -{-  2p.7^  +  (p'  -f  2?i)  x"  + 
2pnx  -f  n^ (2) 

If  therefore  we  should  add  to  both  members  of  eq  (1)  the 
quantity  (p'  -{-  2n)  x'  -j-  2pnx  -j-  ''^^  the  first  member  would 
be  a  perfect  square. 

The  second  member  becomes 
(p^  -}-  2n  -{-  q)  of  -j-  {2xn  -j-  r)  a?  -[-  {n''  +  «);  which,  being 
in  reality  a  trinomial  arranged  according  to  the  descending 
powers  of  x,  will  be  a  perfect  square  if  4  times  the  product 
of  the  extreme  terms  =  the  square   of   the   middle  term; 


78  PRINCIPLES  OF  ALGEBRA. 

that  is  if  4  {p^  +  %i  +  q)  {ii^  -f  s)  =  {2pn  +  rf ;  leaving  off 
for  the  moment  the  powers  of  x.  Performing  the  opera- 
tions indicated,  transferring  all  the  terms  to  the  first  mem- 
ber and  arranging  according  to  the  undetermined  quantit}^ 
n  we  get 

8/i^  4-  Iqri'  +  (8.S'  —  Apr)  n  +  Aqs  +  ipK^  —  r^  =  0.  .  .  .(3); 

an  equation  of  the  3d  degree  with  respect  to  n.  If  we  can 
solve  this  conditional  cubic  equation  and  get  a  value  for  ??, 
this  value  of  n  will  be  what  is  required  to  make  the  trino- 
mial (7/  -f-  2/1  -f  q)  a;^  -f  (2p?2^  -j-  r)  .r  -{-  (n^  -[-  s)  a  perfect 
square.  The  quantity  involving  n  which  w^as  added  to  both 
members  made  the  first  member  a  perfect  square,  and  with 
the  value  of  n  (which  we  still  call  n)  suppose!  to  have  been 
found  from  the  cubic  (3)  both  members  will  be  exact 
squares : 

(^'  -f  2^x  +  ny=  {p"  J-  %i  -f-  q)  .r'  -[-  (2/)/i  -f  r)x^{n'^s); 
taking  their  square  roots  we  have 

x^  -{-  px  -[-  n=  ±  \_\/x>^  -f-  2?i  -j-  7.  ^  -f  -\/n^  -f  ^J  when  the 
middle  term  of  the  trinomial  which  makes  the  second  mem- 
ber is  positive;  and  when  that  middle  term  is  negative  we 
have  x\-\-  px  -{^  7i  =  ± \j\/if  -{-  %i  -\-  q.x  —  \/ ii"  -f  s\ . 

In  either  case  we  have  two  equations  of  the  2d  degree 
and  will  get  4  values  of  x,  wdiich  are  the  4  roots  of  the 
original  equation. 

Art.  90.  This  method  can  be  applied  only  to  those 
equations  of  the  4th  degree  wliich  have  two  imaginary  and 
two  real  roots.  For  let  us  suppose  the  roots  to  be  repre- 
sented by  a,  h,  c  and  d:  the  product  of  two  of  them,  say 
ah,  Yvill  equal  the  absolute  term  of  one  of  the  quadratics, 
giving  ah  =  11  —  y  n'  -[-  s;  .  •  .  n  —  ah  =  \/n'  +  .s  squaring 
which  we  get 

n'  —  2'ihn  -f-  a'b'  =  n'  -^  r  .  •  .  —  2ahn  -f  a'h'  =  s, 
but  as  s,  the  absolute  term  of  the  proposed  equation,  is  equal 


waring's  method.  79 

to  —  abed,  we  have  — 2ab)i  -{-  a^b^  =  — abed  .-.  ab  — 2/i  = 

—  cd  .'.  n  = o •     Similarly  the  other  two  values  of  n 

ac  4-  bd       ^           ad  4-  be 
are  n  = ^ and  n  = j^— 

If  now  all  the  roots  of  tlie  bi-quadratic,  or  equation  of  the 
4tth  degree,  a,  b,  e  and  d  should  be  real,  the  values  of  n,  or 
the  roots  of  the  equaJon  of  the  3d  degree  would  all  three 
be  real  and  we  could  not  solve  it. 

Again  if  a,  b,  e,  d  should  all  be  imaginary,  their  pro- 
ducts, two  and  two,  being  real,  the  3  values  of  n  would  all 
be  real  and  we  could  noi:  solve.  But  if  two  of  the  roots  a^ 
b,  c  and  d  are  real  and  two  imaginary  there  will  be  one 
root  of  the  cubic  real  and  two  imaginary  and  then  Cardan's 
Formula  would  apply.     Thus  suppose  that  a  and  c   were 

n    ,  .         •  ,,         ,  ^<^'  +  ^^  ,  -.   , 

real,  b  and  d  imaginary;  the  value  n  =  — ~o~~~  wotild  be 

real  because  we  should  have  the  product  of  two  real  -|- 
the  23roduct  two  imaginary  quantities,  and  the  sum  would 
be  real.     The  other  roots  would  evidently  be  imaginary. 

EXAMPLE. 

X*  —  6x^  +  5x^  -f-  2a;  —  10  ==  0;  by  comparing  this  with 
the  formula  equation  we  find 

2p  =  —  6orp=  —  3 

q  =  -5 

r  =  — 2 

s  =     10  and  equation  (3)    then  becomes 
8m'  —  20n'  +  56n  +   156  ^-  0,  which  divided  by  4  gives 
2ii^  —  5/i^  -\-  lin  -f-  39  =  0.     On  solving  this  we  find  one 

root  n  —  —  _     Hence 

(.^^  -Sx-^^^'  +  lx  +^|  .-.  .X'  -  3^  -|  =  ±  (^+^j 

.• .  x^  —  Ax  5  and  also,  the  other  quadratic,  x'^  —  2x  = 
—  2:     From  these  we  get  x  =  —  1,  5,  1  -[-  y/ — 1  and  1  — 


80  PEINCIPLES   OF   ALGEBRA. 


The  solution  of  the  intermediate  cubic  results  thus: 
n^  —  -  71^  -f  7?i  +  y  =  0;  make  n  =  |  .♦.  ^/^  —5y''  -f   28y 
+  156  =  0;  make  2/  =  ^-l-|  .-.^3  +  |^  ^  +  ^-^?  =  0. 

Make  t==~.-.  s'  -\-  Ills  +  5222  =  0.  In  this  last  equation 
o 

for  the  purpose  of  applying  Cardan's  formula  we  note  p  = 
177  and  q  =  5222,  hence  ^  =  205379  and  ^.  --=  6817321 
and  the  algebraic  sum  of  the  two  cube  roots  in  the  formula 
=  -14^..   ...  ^  =  3=-^-and^=/  +  3^--- 

n  0/ 

+ 


n 


5 
3' 

-  —  3 

.'.  n  — 

y 

2 

3 
~        2* 

EXAMPLE  2. 

• 

Find  the  i 

•oots  of 

x' 

—^r^— 17^'^— 3a;— 60 

=  0 

^^^  ~ 

17 

2* 

Ans.    —4,    5,    ,/ 

EXAMPLE  3. 

-3,  - 

-V— 

-3. 

Fin 

d  the  roots  of 

x' 

+7.r^— 33^-^+107iP- 

-154- 

=  0. 

=  - 

15 
2* 

Ans.    2,  —11,  l+i/- 

-6,  1- 

-/= 

-6. 

CHAPTER    VII. 

Occasional  Solution  of  Higher  Equations. 

Art.  91.  Beyond  equations  of  the  fourth  degree  there 
are  no  direct  methods  for  exact  solution;  and  as  has  been 
seen,  the  existing  methods  do  not  apply  to  all  equations  of 
the  third  and  fourth  degrees .  But  when  equations  of  any 
degree 


OCCASIONAL  SOLUTION  OF  HIGHER  EQUATIONS.  81 

HAVE    EQUAL    ROOTS 

those  equal  roots  maybe  discovered  by  Art.  53,  and  divided 

out,  thus  reducing  the  degree  of  the  equation.  If  the  re- 
duced degree  is  the  first  or  second,  the  remaining  root  or 
roots  may  be  directly  found;  if  the  reduced  degree  is  the 
third  and  the  equation  has  two  imaginary  roots,  Cardan's 
Formula  will  apply  and  tlie  equation  may  be  directly 
solved;  also  when  the  resulting  equation  is  of  the  fourth 
degree  and  has  two,  and  only  two,  imaginary  roots,  the 
equation  may  be  exactly  solved. 

Art.    92.     Also  when  an  equation 

OF    THE    THIRD    DEGREE    HAS    COEFFICIENTS    WITH    CERTAIN 
SPECIAL    RELATIONS, 

the  roots  may  be  found  exactly. 

Suj)pose  the  equation  to  be  of  the  form 
a?  J^  dx""  -{-  kx  =  q  ....(1) 

Jf  we  had  a  cubic  equation  of  the  following  form 

x^  +  ^px^  +  ^p^x  ==q  .,..{2) 

it  is  evident  upon  inspection  that  if  we  add  j/  to  the  first 
member  it  would  become  a  perfect  cube.  Adding  p^  then 
to  both  members  we  have 

or"  +  dpx^  +  dp'x  +  /  =  (/  +  p^  . .  .(3)  or 

{x  -\-  pY  =  q  -\-  p^  whence  x  =  —  p  -j-  ^^q-\-p^  •  •  •  •  (4) 

Comparing  equations  (1)  and  (2)  we  see  that  d  =  3p  and 

d  k 

k  =  3//,  whence p  ^  ^  and p^  =  -.     Squaring  both  meni- 
o  o 

d  d^  d^ 

bers  of  p  =  -we  havep'^  =  —  .•.  by  addition,  2p^  =  —  + 
o  a  y 


3  =  -3-    .-.p^+^Z-yg- (5) 

From   this   last  formula  the  value  of  p  may  be  found, 
which  is  necessary  to  transform  equation  (1)  into  equation 

11 


82 


PKINCIPLES  OF  ALGEBBA. 


(2)  and  which  in  (4)  will  give  a  root  of   the  proposed  equa- 
tion . 

But  this  is  on  condition  that  p  have  the  same  value  in 
d  =  ^p  and  ^'  =  3^^  that  is,  that  d?  =  ^k .  When  the  coeffi- 
cients of  x^  and  x  are  such  that  the  square  of  the  coefficient 
of  x"^  is  equal  to  three  times  that  of  x,  this  method  will  ap- 

EXAMPLE  1. 

a^^l^x'^^lbx  =  —125. 
Here   d  =  15  and  d'  =  225,  and  3^-  =  3X75  =  225;  and 


/225-f225 


p  =^ -^ =5,  and  x  =  —5+i3/— 125+125   =  —  5, 

and  this  root  divided  out  gives  the  quadratic  ^--f  10a;-j-25= 
0,  of  which  the  roots  are  — 5  and  — 5.  All  three  of  the 
roots  are  real  and  equal  in  this  equation . 

EXAMPLE  2. 

x'-{-lBx'-i-15x  =  218. 
d 


Here  p  =^-  =  5,  and  x  =  — ■p-fif^'^H-p^  =■  — 5-[- 
o 

#'218  4-125  =  —5+1^/243  =  — 5-[-7  =--  2. 

The  other  roots  are : 


—17+  V  —147      ^  —17—1   —147 

X  = —^ —  and  X  = 


EXAMPLE  3. 

7 


^+-H3     3 


Aus.    X  =  ly   X  ^=  — '^~^\l — Q'  *^  ^^  — ^ — \/' 


It  will  be  observed  that  when  this  relation  holds  betvv'een 
the  coefficients,  if  the  second  term  be  made  to  disappear 
the  third  wall  disappear  also .  Further,  that  when  applica- 
ble it  is  so  without  respect  to  the  nature  of  the  roots,  as 
imaginary  or  not.     If,  then,  the  intermediate  cubic  in  the 


OCCASIONAL  SOLUTION  OF  HIGHER  EQUATIONS.  83 

solution  of  an  equation  of  the  fourth  degree  should  be  of 
the  class  just  described,  it  would  enable  us  to  solve  that 
equation  without  respect  to  the  nature  of  its  roots,  widen- 
ing by  so  much  the  field  of  apj)lication  of  Waring's  and 
Descartes'  methods. 


WHOLE-NUMBER   ROOTS. 

Art.  93.  If  an  equation  has  been  placed  in  the  reduced 
form,  X'"  +  Fx'"-'  +  Qx'"-'  _|-  . . .  .  _j_  7-^  _|_  C^=  0,  (1)  ft 
cannot  have  any  roots  which  are  fractions. 

By  fractions  is  here  meant  irreducible  fractions,  or  frac- 
tions whereof  the  two  terms  are  prime  with  respect  to  each 
other.  Such  fractions  containing  "one  or  more  of  the 
equal  parts  of  unity"  are  commensurable  with  unity. 
Whole  nnmbers  are,  of  course,  commensurable  with  unity. 
See  Art.  2G. 

Art.  94.  Consequently,  when  we  find  no  whole  num- 
ber among  the  roots  of  an  equation  of  the  reduced  form, 
since  we  already  know  that  there  are  no  fractions  among 
them,  we  know  that  none  of  the  rools  are  commensurable 
with  unity.  3-|-|/2,  f/1  ^  are  specimens  of  quantities  not 
commensurable  with  unity.  Imaginary  quantities  are 
never   commensurable  with   unity. 

Art.  95.  The  absolute  term  of  the  equation  will  con- 
tain as  divisors  all  the  roots,  whether  wh^le  numbers  or 
not;  but  it  will  usually  contain  many  other  divisors  besides 
the  roots.  We  could  scarcely  hope  to  find  what  the  in- 
commensurable divisors  of  any  absolute  term  are,  bnt  we 
can  more  easily  discover  those  which  are  whole  numbers; 
and  of  these,  taking  those  lying  between  a  superior  limit 
of  the  positive  roots,  L,  and  a  numerically  superior  limit  of 
the  negative  roots,  — U\  we  can  discover  by  trial  which 
among  them  are  roots. 

But  thp  labor  of  these  substitutions  may  be  much  short- 
ened by  the  results  of  the  following  investigation. 


84  PRINCIPLES   OF  ALGEBRA. 

Art.    96.     Let  a,  a  whole  number,  be  a  root.     Then 

and  transposing  to  the  second  member  all  the  terms  except 
U,  and  dividing  by  a,  we  have: 

-  =  —a^-^—Pa^-^—  ....  —Ea'—Sa—T (1) 

Since  the  second  member  contains  none  but  whole  num- 
bers, it  is  entire,  and  therefore  —  is  entire,  which  is  merelv 

a 

confirmatory  of  what  we  already  knew.  Now  transposing 
T  to  the  first  member,  and  dividing  by  a,  we  obtain : 

— h  2'=  — a— ^— Pa— •?— .  . .  ,—Ea—S, 
a 

and  as  the  second  member  is  entire,  we  see  that  the  quotient 
of  the  absolute  term  divided  by  the  whole  number  root,  plus  the 
coefficient  ofx  is  also  exactly  divisible  by  that  root. 

For  — j-  T  substitute  T\  transpose  S  and  divide  by  a  as 
a 

before,  and  the  result  is: 

— ^^  =  —a-^-3—Pa^-4—  ....  —Qa—R, 
a 

and  as  this  second  member  is  entire,  we  see  that  the  former 
quotient  plus  the  coefficient  of  x^  is  exactly  divisible  by  the  root. 

Now  making  — ^^^^—  =  S\  transposing  — B  and  dividing 

by  a  as  before,  ijiere  results: 

0/_[_     73 
^--L^  =  —a^n-4_p^.n-s_ _Q 

a 

a  whole   number.     Therefore  the  last  preceding  coefficient 

plus  the  coefficient  of  a^  is  exactly  divisible  by  the  root. 

Proceeding  in   the  same  manner,  when  we  shall  have 

transposed  all  the  terms  save  two,  we  shall  have  an  equa- 

O' 
tion  like  — =  — a — P  =  a  whole   number.       Transposing 

~4-P      P' 
—P  and  dividing  as  before:  a         =  —  =  — 1.  This  shows 

a 


OCCASIONAL  SOLrTION  OF  HIGHER  EQUATIONS.  85 

that  the  last  of  the  quotients  (which  is  forn^d  when  the 
coefficient  of  the  second  term  is  transposed)  is  — 1 .  Every 
divisor  of  the  absolute  term  which  will  stand  all  of  these 
successive  tests  is  a  root,  and  as  it  is  supposed  that  we  will 
try  only  those  divisors  which  are  whole  numbers,  we  will 
discover  all  the  whole-number  roots . 

Having  found  them  we  divide  them  out,  as  in  the  case  of 
equal  roots,  and  solve,  if  possible,  the  resulting  equation. 

We  may  form  a  table  at  the  heads  of  the  vertical  columns 
of  which  are  placed  all  the  entire  divisors  which  lie  between 
the  upper  and  lower  limits,  and  then  make  a  simultaneous 
trial  of  them  all;  rejecting  all  which  in  any  of  the  succes- 
sive divisicHis  give  quotients  not  entire;  that  is,  any  which 
fail  to  stand  all  the  tests. 

Having  formed  the  table  by  writing  the  whole-number 
divisors  between  the  limits  in  a  horizontal  row  proceed  by 
the 

RULE 

Divide  the  absolute  term  by  each  divisor  setting  the  quotient 
immediately  beneath  the  divisor.  Form  new  dividends  by  ctdd^ 
ing  the  coefficient  of  x  to  the  quotients.  Divide  these  by  the 
numbers  on  trial,  setting  the  quotients  immediately  beneath  the 
dividends.  Form  new  dividends  by  increasing  the  last  quoti- 
ents by  adding  to  them  the  coefficient  of  x^.  So  proceed,  al- 
ways forming  new  dividends  by  the  addition  to  the  last  quotients 
of  the  next  succeeding  coefficient  towards  the  first,  and  re- 
jecting any  divisor  which  at  any  stage  gives  a  fractional  quoti- 
ent. All  those  which  finally  give  a  quotient  which  is  minus 
unify  are  roots . 

EXAMPLE   1. 

Find  the  entire  roots  of  the  equation : 

9.r'-|-30^^-f22^*+10.x'+17;r'— 20^+4=0. 

,  ,  lO-r^     22  .     10  _     17  ,     20       4'  ,         .  y 

^'+^-+9-^  +  :^^+ 9-^  —  9 '^+9=0;  make  ^=3.-. 

2/6_|_io?/5_|.22?/*+302/'r|-153?/'— 540i/+324=0. 

Here  L  =  4:  and  —  U'  =  —  31 
And  the  divisors  (which  are  whole  numbers)  of  the  absolute 


86 


PRINCIPLES    OF   ALGEBRA. 


term  324,  are  4,  3,  2,  1, 
—18  and  —27. 


-2,  —3,  —4,  —6,  —9, 


-12, 


3 

.108 

2 
162 

1 
324 

— 1 
-324 

-Ti 

—3 
—108 
—648 
+216 

—4 

—81 

-6 
-54 

—9 
—36 

-12 

-27 

-18 
—18 

—27 
—12 

—432 
—144 

—378 
-189 

-216 
—216 

—864 

+864 

1017 

—1017 

—708 
+354 

—621 
+  155  J 

—594 
+99 

—576 
+64 

-567 

—558 
+31 

184 
—102 

9 

-552 
+20'-^ 

9 
+3 

—36 
—18 

—63 
—63 

507 
-253J 

369 
—123 

252 
—42 

217 
_24l 

27 

33 
+11 

12 

+6 

—33 
—33 
—11 
—11 

—987 
+987 

-93 

+31 

53 

-171 

— 1-Z 

+  2 

24 

—4 

33 
+11 

28 
+  14 

1009 
—1009 

21 

+7 

24 
+  12 

-1 
— 1 

—999 
+999 

+6 

— 1 

The  onl^r  two  divisors  giving  a  final  result  of  —  1  are 

y  1 

-}-  1  and  —  6    and  placing  these  in  ^  =  ;^  we  have  x  =  - 

o  o 

and  X  =  —  2  which  will  satisfy  the  equation  j^roposed. 
2x'  —  15^=^  +  8x'  +  GSx  +  48  =  0. 


2.     X 
3 


Ans.     —  2,  —  2,  —  1,  3  and  4. 

.^4_5^3_j_  25^  —  21  =  0. 


•      ^   ,    1  +  1/29        ,1 
Ans.     3,1, — —^ and— 


l/29 


SOLUTION  OF  RECURRING  EQUATIONS. 

Art.  97.     These   equations  have   their   coefficients  re- 
cur   when     counted    from    the    first    and    last.       Their 

roots  are  of  the  form  «,  -,  b,  -.  etc. 
a        b 

The  student  should  now  carefully  review  Articles  41  to 
43,  inclusive. 

Binomial   equations  of   the  form   oi?-\-l  =  0,   x'^-\-l  =  0, 
X* — 1  =  0,    etc.,  are  recurring  equations. 

1.     Take  x^ — 1  ^0;   we  know  that  -|-1  is  a  root  of  this 


equation;   dividing  it  out,  we  get 


X'- 


x^-\-x-]-l,  and 


the  roots  of  x^-^x-{-l  ==  0  are ^ and  — — ^ . 


OCCASIONAL  SOLUTION  OF  HIGHER  EQUATIONS.  87 

2.     Take   ,i^~{rl  =^0;   we  know  that  — 1  is  a  root  of  this 
equation,  and  dividing  it  out  we  get    ~|~    ■  =  on^ — '^'  fl;  the 

roots  ot  r^ — i>t*  -1  =  0   are ^— and 7= 


3.  Take  a:^ — 1  =  0;  we  know  that  this  is  composed  of 
two  factors  of  the  second  degree,  to  wit,  x'^ — 1  and  x^-\-l; 
hence  placing  these  equal  to  zero  and  solving,  we  get  the 
four  roots,    -f  1,    — 1,    -{-V—l  and  —  l/^. 

4.  Take  x'—l  =  0;  this  is  (.r^  fl)(.r'— 1)  =  0,  giving 
cases  1  and  2. 

5.  j:,-5_|_;i^  ^  0  is(x-{~l){x'—x^-\-.jc^—x^~l)  =  0,  giving^ 
-|-1  -^^  0  and  X'* — a^-\-x- — ^+1  =  0.  This  last  is  a  recur- 
ring equation  of  the  fourth  degree. 

But  before  examining ^*-fl  =^i  .x'*'-l-l  =  0,  etc.,  we  will 
demonstrate  the  following  principle: 

Art.  98.  Every  recurring  equation  of  the  fourth  and 
higher  even  degrees  may  he  solved  by  using  one  of  a  degree  half 
as  high . 

Suppose  we  had  a  recurring  equation  of  the  fourth  de- 
gree: 

x'-{-Px'^Qx'+Fx+l  =:  0, 

and  that  the  roots  were  a,  -,  b  and  j  ;  then  the  factors  of 

the  first  degrfee  would  be  x — a,   x ,  x — b  and  x — -  ,  and 

a  0 

the  quadratic  factors  would  be  x"^ — K^+^/i-  +  1    and 

x^ —  \b-\--jx-[-l .     Put  a-\ —  =  k  and  b-{-~  =zl;  then  we  have 

x^^kx-{-l  and  x"^ — lx-\-l.  These  multiplied  together  give 
the  equation  of  the  fourth  degree  in  w^hich  x  enters  as  a 
factor  four  times,  while  k  enters  only  twice;  therefore  what- 
ever equation  we  may  obtain  for  the  value  of  k,  from  or  by 


88  PRINCIPLES  OF  ALGEBRA. 

means  of  the  original  equation,  will  be  only  of  the  second 
degree .  If  we  multiply  the  quadratic  factors  involving  k 
and  /,  and  place  the  product  equal  to  the  original  equation, 
we  will  form  an  identical  equation,  and  equating  the  cor- 
responding coefficients,  we  could  determine  the  values  of  k 
and  /;  and  it  would  be  found  that  none  of  the  subordinate 
equations. would  be  above  the  second  degree. 

Again,  suppose  the  recurring  equation  to  be  of  the  sixth 

decree  and  put  k  =  «+-,    I  =  &+,    and  h  ^=  c-\-~  .       The 

°  ^  a  b  c 

quadratic  factors  x^ — ^j?+1j  ^~ — ?J^+1  and  x^ — }iX'\-l  mul- 
tiplied together  give  x^—{k^l^h)r'-^{kl^kh^lh-{-Z)x'— 
{klh^2k^  2/+  2h)x^^  {kl-\-  kh-\-  lh^^y—{k^  Z-f-  h)x^  1 . 

Put  this  equal  to  the  first  member  of  original  equation, 
supposed  to  be  a^^Px^-{-Qx^-^Rx^^Qx^-\-Fx-\-\^  and  equat- 
ing the  coefficients,  we  get: 

Q  =  kl-^kh-f-lh-^-S; 
B  =  —{klh^2k^2l-i~2h); 
in  which,  since  P,  Q,  and  E  are  known  numbers,  we  have 
to  determine  the  three  unknown  quantities,  k,  I  and  h  from 
the  three  equations,  one  being  of  the  first,  one  of  the  sec- 
ond and  one  of  the  third  degree.  The  resulting  equation 
would  therefore  be  of  the  third  degree,  one-half  the  degree 
of  the  original  equation. 

A  similar  investigation  would  evidently  show  a  similar 
result  for  any  equation  of  a  higher  and  even  degree . 

Returning  to  Example  5 — we  there  saw  that  one  of  the 
roots  was  — 1,  and  when  this  was  divided  out  there  resulted 
the  recurring  equation  x^ — ^-f  ^'^ — x-{-l  =^  0  =x* — (k-\-l)x'^ 
-i-{kl^2)x'—{k^l)x-^l.     From  this  1  =  k-{-l  and  also  1  = 

kl4-2   .  • .    k'—k  =  1  and  ^  =  ^J^.     But  as  a+-  =  k. 


H 


l±|/5±i/— lQ-f2|/5 
4 


OCCASIONAL  SOLUTION  OF  HIGHER  EQUATIONS. 


89 


and  either  of  these  four  roots  when  substituted  in  x^-\;-l  = 
0,  will  satisfy  it. 


For  instance  take  the  first: 


1  +  v/54-l/-10+2i/5   ^^^ 


raise  it  to  the  fifth  power.  4'^  =  1024.  In  the  numerator 
place  1  +  /5  ^  a  and  i/— 10+2v/5  =  d.  Then  (c-^df  = 
c^+5cV«+10c^^*^+10c=(f+5ccZ*-l-cf   and 


4- 

o 


or 


rn 

o 

o 

or 

o 

^co 

o^ 

S 

% 

a: 

a] 

%. 

i 

II 

if 

II 

li 

rf^ 

lo 

h- 1 

o 

<^ 

-q 

o 

1 

r 

+ 

n^ 

tf^ 

o 

00 

00 

o 

o 

p 

.^ 

\ 

-x 

en 

o 

2 
\ 

1 

o 

+ 

bO 

OX 

1 
3 

\ 

en 

\ 
1 

M 

O 

+ 
err 

CI 

+ 

bO 

§ 

I 

o 

+ 

or 
+ 

1 

O 
+ 

OX 

OX 

— 1024,  and  this  divided  by  the  fifth  power  of  the  denom- 
inator gives  — 1.  which  satisfies  the  equation.  In  a  similar 
manner  other  recurring  equations,  like  a^-\-l  =  0  and 
x"'-\-l  =  0,  may  be  solved  whenever  we  caij  solve  an  equa- 

12 


90  PRINCIPLES  OF  ALGEBRA. 

tion  of   half   the  degree,  and  this  is  true  of   all  equations 
which  are  recurring,  whether  binomial  or  not. 


EXPONENTIAL    EQUATIONS. 

Art.  99.  Exponential  equations,  or  such  as  have  the 
unknown  quantity  as  an  exponent,  sometimes,  but  rarely, 
admit  of  exact  solution.  It  is  assumed  that  the  student  is 
familiar  with  the  solution  of  exponential  equations  by  con- 
tinued fractions  and  logarithms. 

These  equations  do  not  fall  within  the  class  of  algebraic 
equations,  but  of  transcendental  equations. 


CHAPTEK     VIII. 

Approximate  Solutions  of  Higher  Numerical  Equations. 

Art.  100.  "When  it  is  not  practicable  to  solve  an  equa- 
tion by  any  of  the  modes  which  have  been  discussed,  we 
must  rest  content  with  an  approximation  to  the  roots.  For- 
tunately this  can  be  had  closely  enough,  for  practical  pur- 
poses, by  the  methods  which  we  now  propose  to  examine. 

We  have  seen  that  when  an  equation  has  some  equal 
roots,  or  some  that  are  whole  numbers,  we  may  discover 
them  and  divide  them  out,  and  reduce  the  degree  of  the 
equation;  if  it  is  a  recurring  equation,  we  shall  have  to 
solve  one  only  half  as  high  in  degree;  and  in  short,  if  by 
trial,  chance  or  in  any  other  way,  we  can  discover  one  or 
more  roots,  we  would  immediately  depress  the  degree.  If 
after  all  it  is  of  the  fifth  or  higher  degree,  we  can  only 
approximate,  and  so  likewise  with  those  of  the  third  and 
fourth  degrees  when  they  do  not  happen  to  have  two  imagi- 
nary roots  and  no  more;  unless  there  is  a  peculiar  relation 
among  the  coefficients  such  as  was  examined  in  Art.  92. 

Horner's  method. 

Art.  101.  This  method  was  first  published  in  1819.  It 
is  the  invention  of  W.  G.  Horner,  and  is  regarded  by  most 


APPROXIMATE  SOLUTIONS.  91 

mathematicians  as  the  most  satisfactory  mode  of  approxi- 
mating to  the  real  and  incommensurable  roots  of  an  equa- 
tion having  numerical  coefficients.  The  method  is  as 
follows : 

1st.  Having  found  by  Sturm's  Theorem,  or  otherwise, 
the  whole-number  part  of  a  root,  and  still  better,  having 
found  in  addition  one  or  more  of  the  figures  in  the  decimal 
part,  to  transform  the  original  equation  into  another  whose 
roots  shall  be  less  by  the  part  already  found . 

2d.  To  obtain  the  next  figure  of  the  root  by  dividing  the 
absolute  term  by  the  next  preceding  coefficient  and  taking 
the  first  figure  of  the  quotient  for  the  required  figure . 

3d.  Then  to  transform  this  equation  into  another  whose 
roots  shall  be  less  by  the  decimal  figure  last  obtained;  to 
divide  the  absolute  term  of  this  equation  by  the  coefficient 
which  immediately  precedes  it,  and  take  the  first  figure  of 
the  quotient  for  the  next  figure  of  the  root. 

4th.  Again  transform  the  equation  into  another  of 
which  the  roots  shall  be  less  by  the  decimal  figure  last  ob- 
tained, divide  the  last  coefficient  by  the  one  immediately 
preceding  for  the  next  figure  of  the  root,  and  so  continue 
till  the  desired  number  of  places  in  the  approximate  root 
shall  be  found. 

In  this  way  we  find  the  real  positive  roots,  and  if  there 
are  any  which  are  negative,  obtain  them  approximately  by 
changing  the  alternate  signs  of  the  proposed  equation, 
which  will  make  the  roots  now  being  sought  all  positive, 
and  proceed  as  before. 

If  preferred,  when  one  or  more  roots  have  been  found, 
they  may  be  divided  out  and  the  degree  of  the  equation 
reduced. 

Demonstration . 

Art.  102.  When  an  equation  has  been  transformed 
into  another  of  which  the  roots  are  less  by  the  whole- 
number  part  of  the  original  root,  and  still  more  if  they  are 
less  by  the  whole-number  part  and  one  or  more  figures  of 


92 


PRINCIPLES  OF  ALGEBRA. 


the  decimal  part,  the  remainder  of  the  root,  that  is,  the 
value  of  the  unkno\Yn  quantit}'  in  the  transformed  equa- 
tion, is  a  very  small  quantity  indeed.  Therefore  its  second 
and  higher  jDowers  may  be  neglected  in  comparison  with 
itself,  and  the  first  member  of  the^  transformed  equation, 
which  will  be  of  the  form 

may  be,  without  appreciable  error,  taken  to  be 


T-\-  U 


0; 


and  the  first  figure  of  the  quotient  (and  j)erhaps  more)  will 
be  the  initial  figure  or  figures  of  the  true  value  of  u,  and 
will  therefore  be  the  next  required  figure  in  the  value  of 
the   original  unknown  quantity. 

Let  it  be  required  to  find  the  approximate   roots  of   the 
roots  of  the  equation : 

^4_8^-}-14a;2_]_4^_g  ^  0. 

Sturm's  functions  of  this,  when  reduced  to  their  simplest 
form,  are: 


1 
8 

+ 

+ 

+ 
4 

+ 
8 

+ 

+ 

+ 

+ 

+ 

0 

+ 

+ 

+ 

3 

1 
+ 

+ 

+ 
4 

+ 

+ 
+ 

+ 
2 

+ 
to 

+ 

+ 
+ 
2 

+ 

± 

+ 
+ 
1 

+ 

+ 

+ 
+ 
1 

+ 

OS 

+ 
+ 
+ 
+ 
+ 
0 

V,=  hx'—Vlx^Q 
V,  =  76^—103 
Vi=^-\-  number 

Variations 


Since  the  number  of  variations  lost  between  — 1  and  -}-6 
is  the  same  as  between  — oo  and  -}-oo  ,  all  the  roots  are  real 
and  comprised  between  — 1  and  6. 

Between  — 1  and  0  there  is  4 — 3  =  1  variation  lost; 
there  is  one  negative  root,  then,  between  0  and  — 1,  and 
since  it  is  numerically  less  than  — 1,  we  substitute  in  V, 
Fi,  F2,  etc.,  in  succession,  — .1,  — .2,  — .3,  etc.,  until  there 


APPROXIMATE    SOLUTIONS.  93 

is  a  gain  of  variation,  and  the  last  preceding  number  will 
be  the  first  figure  of  the  root .  This  root  in  the  exam j)le  is 
found  between  — .7  ond  — .8;  therefore  .7  is  the  first  fig- 
ure of  th6  negative  root. 

0  gives  3  variations,  and  -\-l  gives  2;  hence  there  is  a 
positive  root  which  is  a  decimal  fraction.  By  the  succes- 
sive substitution  of  .1,  .2,  .3,  etc.,  its  first  figure  is  found 
to  be  .7. 

2  gives  2  variations  and  3  gives  1;  hence  there  is  a  root 
whose  first  figure  is  2;  and  as  5  gives  1  variation  and  6 
gives  none,  there  is  a  root  whose  first  figure  is  5.  This  last 
fact  may  be  known  at  once,  because  5  in  F  gives  a  negative 
result  and  6  in  F  gives  a  positive  result.  There  is  there- 
fore one  real  root  between  them,  or  else  some  other  odd 
number  of  roots . 

Let  us  now  proceed  after  Horner's  manner  to  find  this 
root,  whose  whole-number  part  is  5.  The  coefficients  of 
the  original  equation  are: 


—8 

+5 

+14 
—15 

+4 
5 

—8    1  5 
—5 

3 

+5 

—1 

+10 

—1 

+45 

,-13 

+2 
+5 

+7 
+5 

+9 
+35 

,+44 

,+44 

1  ,+12; 

and  13^-44  :^=  .2 ;   .2  is  the  next  figure  in  the  root,  and 

the  coefficients  of  the  transformed  equation,  that  is,  one 
whose  roots  are  less  than  those  of  the  original  equation  by 
5,  are:  1,  +12,  -[-44,  -f  44,  —13.  It  will  be  observed  that 
they  run  in  a  diagonal  line  from  left  to  right  ux)wards  in 
the  calculation  of  the  transformation. 

Now  let  us  get  an  equation  whose  roots  are  less  by  .  2 
than  those  of  the  equation  whose  coefficients  are: 


94  PRINCIPLES    OP    ALGEBRA. 


1  +  12      +44        +44 
0.2   +   2.44  +   9.288 

-13 

+  10.6576 

+  12.2   +46.44  +53.283, 
.2  +   2.48  +   9.781 

,-    2.3424 

+  12.4  +48.92, +  63.072 
.2   +   2.52 

+  12.6, +  51. 44 
.2 

1  +  12.8 

2.3424-^63.272  gives  .03;  hence  3  is  is  the  next  figure  of 
the  root,  and  transforming: 

1  +  12.8     +51.44       +63.072         -2.3424  |  .03 

.03  +      .3849  +   1.554747  +1.93880241 


12.83  +51.8249  +64.626747  ,-.40359759 
.03  +     .3858  +   1.566321 


12.86  +52.2107, +  66.193068 
.03  +     .3867 


12.89, +52.5974 
.03 


1  +  12.92 

.40359759-f-66.193068  gives  .006,  and  6  is  the  next  figure 
of  the  root.     Again  transforming: 

1+12.92    +52.5974      +66.193068       -  .40359759         |  .006 
.006  +     .077556  +     .316049736  +.399054706416 


12.926  +52.674956  +66.509117736,— .004542883584 
.006  +     .077592  +     .316515288 


12.926  +52.752548  +66.825633024 
.006  +     .077628 


12.932,+52-830176 
.006 


1  +  12.938 

and  .004542883584--66.825633   gives  .00006;   hence  06  are 
the  next  two  figures  of  the  root.     Again  transforming: 

1  +  12.944     +52.830176  +66.825633  -^.004542883656  L'^^l- 

.00006+     .0007766436    +     .003169857158616    +.00390972817142951696 


12.94406+52.8309526436    +66.828802857158616  ,—.00063315548457048304 
.00006+     .0007766472     +     .0031699037584992 


12. 94412+52.83172930832,+66. 83197  2760917152 


APPROXIMATE    SOLUTIONS.  95 

As  we  will  not  carry  the  approximation  nearer  than  to  G 
figures,  it  will  not  be  necessary  to  go  further  than  has  been 
done,  since  we  have  now  the  means  of  getting  the  sixth 
figure,  which  is  done  by  dividing  .000533115M845704304 
by  66. 8319727G09171152,  and  we  get  a  number  between  7 
and  8,  but  being  nearer  to  the  latter,  we  put  8  as  the  sixth 
figure  of  the  root,  and  we  have  5.236068.  In  the  same 
manner  we  can  find  the  other  roots. 

Art.  103.  But  as  the  number  of  decimal  places  be- 
comes inconveniently  large,  especially  when  a  considerable 
number  of  decimal  places  are  desired  in  the  root  itself,  we 
must  attempt  some  measure  of  relief.  This  may  be  had  by 
simply  using  no  more  places  of  decimals  than  are  neces- 
sary in  each  stage  of  the  operations. 

Having  decided  on  the  number  of  decimal  places  that 
shall  be  in  the  root,  we  will  remember  that  that  number, 
or  one  or  two  more,  will  be  sufficient  to  have  in  the  divi- 
dends. Also  that  the  number  in  the  dividend  minus  the 
number  of  the  place  of  the  required  figure  of  the  root  at 
any  stage,  will  give  the  number  of  places  that  ought  to 
be  used  in  the  divisor.  Thus,  if  there  are  to  be  6  places  of 
decimals  in  the  approximate  root  and  we  are  multiplying 
by  the  third  figure,  if  the  otheiv  factor,  which  is  the  divi- 
sor, has  3  places,  the  product  will  contain  6  places  and 
give  the  dividend  to  the  necessary  extent.  One  or  two 
places  more  may  well  be  preserved,  and  all  the  others  to 
the  right  dropped;  but  in  multiplications  of  such  reduced 
numbers  we  must  at  the  first  product  on  the  right  hand  in 
every  case,  add  on  the  figure  which  would  have  been  '*  car- 
ried "  there  had  no  figures  been  dropped. 

If  the  number  of  places  of  decimals  can  be  thus  curtailed 
in  the  divisor,  and  since  that  divisor  is  itself  a  product  in 
which  the  last  figure  of  the  root  is  a  factor,  the  coefficient 
preceding  may  be  cut  down  to  a  still  smaller  number  of 
places.  Each  coefficient,  as  we  proceed  from  right  to  left, 
may  have  one  figure  more  dropped  than  was  done  in  the 
case  of  its  immediate  predecessor.     In  this  way  the  coeffi- 


96 


PRINCIPLES    OF    ALGEBRA. 


cients  in  the  left  hand  columns  will  soon  and  successively 
become  constant,  because  all  decimals  would  have  to  be 
rejected,  until  finally  there  may  be  left  only  the  absolute 
term  and  also  the  penultimate  coefficient,  wbich  latter  sim- 
ply loses  one  figure  from  the  right  every  time  a  new  figure 
in  the  root  is  found. 


Art.  104.     This   matter  may  be  illustrated  by  finding 


again  the  root  5.236068: 


1  —8 

+  5 

4-14 
—15 

+4 
—5 

—8      i  5.236068 
—5 

—3 

4-5 

—  1 

4-10 

-1 

4-45 

—13* 

10.6576 

+  2 

+  9 

4-35 

,4-44* 
9.288 

—2.3424* 

1.9388024 

+7 
+5 

,-f44* 
2.44 

53.288 
9.784 

—  .4035975* 
.3990549 

1*4-12* 
0.2 

46.44 

2.48 

63.072* 
1.554747 

—.0045426* 
.0040095 

12.2 
0.2 

48.92 
2.52 

64.626747 
1.566321 

.0005331 

12.4 
.2 

51.44* 
.3849 

66.19306* 
•  .31608 

12.6 
.2 

51.8249 

.3858 

66.50915 
.31656 

1*4-12.8* 
.03 

52.2107 

.3867 

66.826 

12.83 
.03 

52.5974* 

.08 

12.86 
.03 

12.89 
.03 
1  4-12.92* 
.006 

52.68 

.08 

52.76 

- 

12.926 


APPROXIMATE  SOLUTIONS.  97 

The  places  in  which  decimal  figures  have  been  dropped 
off,  and  partial  amends  made  by  increasing  the  last  figure, 
will  be  perceived  upon  inspection . 

Next  let  the  root  of  which  the  first  figure  is  the  whole 
numder  2  be  found. 

1—8  -{-U  -1-4  —  8  I  2.7320508 

4-2  —12  4  16  V 


—6 
2 

2 

—8 

8 
—12 

+  8 

—  7.4599 

—4 
2 

—6  . 
—4 

—  4 

—  6.657 

.5401 
—.50511759 

—2 

.  2 

—10 
.49 

—10.657 
—  5.971 

.03498241 
—.03411504 

0 
0.7 

-9.51 

.98 

—16.628 
—  .209253 

.00086737 
.00085356 

0.7 

.7 

—8.53 
1.47 

—16.837253 
—  .206679 

.00001381 
.0001366 

1.4    _7.06   —17.04393      .0000015 
.7    —  .0849  —  .01359 


2.1  —6.9751  —17.0575 

.7  .0858  —  .0135 

2.8  —6.8893  —17.0711 

.03  .0867 


2.83  —6.802 

.03  .008 

2.86  —6.794 
.03 


2.89 
.03 

2.92 


98 


PRINCIPLES  OF  ALGEBRA. 


The  quotient  of  8  divided  by  — 4  gives  — 2,  which  is 
much  too  small,  as  may  be  found  by  trial;  it  must  be  in- 
creased, and  it  is  found  that  2.7  and  2.8,  when  substituted 
for  X  in  the  first  member,  give  different  signs,  hence  there 
is  a  root  between  them,  and  we  take  .7  for  the  second  fig- 
ure of  the  root.  We  afterwards  proceed  as  usual.  .7  is  a 
quantity  too  great  to  have  its  second  and  higher  powers 
dropped  as  inappreciable. 

The  root  of  which  .7  is  the  first  figure  may  be  found 
thus: 


.7 


-1-14  -{-4  —8  I  .763932 

—5.11  6.223  7.1561 


—7.3 

.7 

8.89 
—4.62 

10.223 

2.989 

— .«439* 
.79211376 

—6.6 

.7 

4.27 
—4.13 

13.212* 
—     .010104 

—.05178624* 
.03951341 

—5.9 

.7 

.14* 
—.3084 

13.201896 
—     .028392 

—.01227283* 
.1184220 

—5.2 
.06 

—.1684 
—.3048 

13.173504* 

—     .002368 

—.0043063* 
.000039472 

—5.14        —.4732        13.171136 
.06        —.3012    —  2412 


.0042668* 


-5.08 
•06 


.7744*   13.15872 
.0149  —     72 


—5.02    —.789    13.158 
.06    —  15   —    07 

1_4.96*   —.804   .  13.15,7,3 


To   obtain  the  fourth  root,  which  is  negative,  we  must 
transform  the  equation  into  another  whose  negative  roots 


APPROXIMATE  SOLUTIONS. 


99 


correspond  to  the  positive  roots  of  tliis,  and  conversely,  by 
changing  the  alternate  signs,  and  we  have  as  follows: 

1_|_8      -^14  —4  —8  I  .7320508 


0.7 

6.09 

14.063 

7.0441 

8.7 

.7 

20.09 
6.58 

10.063 
18.669 

—  .9559, 
--  .89261841 

9.4 
.7 

26.67 
7.07 

28.732, 
1.021947 

—  .06328159, 
.06171029 

10.1 
.7 

33.74, 
.3249 

29.753947 
1.031721 

—  .00157130 
154632 

10.8. 
.03 

34.0649 
.3258 

30.785668, 
.069478 

—  .00002498 
2473 

10.83 
.03 

34.3907 
.3267 

30.85514,6, 
.06952 

25 

10.86 
.03 

34  7174, 
.0218 

30.92467, 
173 

10.89 
.03 

34.739,2 
.022 

30.926,40 
2 

10.92     34.75,1     30.9,2,8 

The  algebraic  sum  of  these  roots  is  equal  to 
8,  the  coefficient  of  the  second  term  with  its 
sign  changed,  as  should  be  the  case: 


5.236068 
2.732050  8 
.763932 

8.732050  8 
—.732050  8 


EXAMPLES. 


1.  Find  one  root  of  ar' — 2x— 5  =  0 . 

2.  Of  x'+lOx'— 24^— 24:0  =  0. 

3.  Of  ar'-{-2x*-\-d.j(^-]-3x'-\-5x— 6^321 


8. 


Ans.  2.0945515. 

Ans.  4.898979. 
0. 

Ans.  8.414455. 


100  principles  of  algebra. 

Newton's  method. 

Art.  105.  In  this  mode  of  ai^proximating  to  the  roots 
of  numerical  equations,  it  will  be  assumed  that  all  the 
roots  which  are  equal  or  which  are  whole  numbers,  have 
been  found  and  divided  out.  Then  having  in  some  way, 
by  Sturm's  Theorem,  by  chance  or  otherwise,  found  a 
a  number,  or  numbers,  which  differ  but  slightly  from  the 
roots  sought,  let  this  approximation,  plus  or  minus  a  new 
unknown  quantity  be  substituted  for  the  original  unknown 
quantity  in  the  first  member  of  the  equation,  and  let  all 
the  indicated  operations  be  performed.  The  new  unknown 
quantity  represents  the  difference  between  the  approximate 
root  which  is  being  tried  and  the  true  root,  and  of  course 
should  be  so  small  a  fraction  that  all  terms  involving  its 
powers  higher  than  the  first  may  be  dro2:)ped  as  inapi3recia- 
ble,  or  at  all  events,  producing  no  serious  error. 

Then  from  this  equation  of  the  first  degree,  find  the  value 
of  this  difference,  and  add  it  to  the  trial  root  when  that  is 
too  small,  or  subtract  it  when  the  trial  root  requires  to  be 
diminished .  We  have  now  an  approximate  root  once  cor- 
rected. But  if  this  be  not  close  enough  to  the  truth,  let  it 
he  used  as  a  trial  root,  precisely  as  before,  and  the  value  of 
a  second  correction  obtained. 

And  this  corrected  root  may  be  used  for  a  third  correc- 
tion, and  so  on  to  any  desired  extent. 
Let  us  take  as  an  example 

o(^-\-Qx'^x  —  10  =  0; 
in  which  it  has  been  found  by  trial  that  1.1  is  an  apj^roxi- 
mate  root,  being  somewhat  too  small .    Let  u  =  the  differ- 
ence between  1.1  and  the  true  root;  we  will  then  have  x=^ 
tt-f-1.1.     For  a  moment  let  1.1  be  represented  by  r     .  • . 

x  =  r^u)  x'=r'-ir2rit-^u';    x'^7''-i-dr'u-\-^rit'^u'; 
and  we  have : 

a^  =  u^-^Su^r-\-3ur^-\-  r^ 

-f  6it?'  =-         6it'  +12ur-[-6?-' 

-\-x  =  u-\-r 

—10  =  —10 


0; 


APPEOXIMATE  SOLUTIONS.  101 

and  dropping  the  terms  involving  u^  and  u^,  we  have: 

{Sr'  +  12r  -f-  1)  ii  -[-  7^  -]-  Gr'  -[-  r  —  10  r^  0,  whence  u  = 

10  _  r  —  Gi''  —  r^ 

— Q  2  _i  ~fo  "ITi —  ^^^  restoring  1.1  in  place  of  r  we  have 

10  —  1.1   —  7.26  —     1.331  ^^rr^^r.^ 

u  =-  17^3 ^  .0173303.  .ajid.x  =  r-{- 

u  =  1.1173303.     Now  if  we  substitute  this  value  of   the 
root  for  r  in  the  formula  above  we  would  get  a  second  val- 
ue of  1^  or  a  second  correction.    But  taking  1.1173  as  being 
one  root  and  dividing  it  out  we  get 
x'  +  7.1173j;  +  8.95215929  =  0;  and  solving  this 
x  =  —  3.55865  ±:  j/ir"8;952r592~9  -f  (—  3.558657T\ 
the  other  two  roots  are  —  5  .  48526 

—  1  .  63204   to   which 
adding  the  first  +  1  .  11730 


—  6  .  00000  which 

is  the  coefficient  of  the  2d  term  with  its  sign  changed  as  it 
ought  to  be.  The  product  of  these  roots  which  ought  to  be 
10  is  10.002252  -]-....      showing  a  good  approximation. 

EXAMPLE    2. 

Find  a  root  of  x^  -\-  x''  -{-  .x  —  100  =  0.  An  approximate 
root  is  4.2,  The  first  correction  by  Newton's  Method  gives 
4 .265 .  .  which  is  a  little  too  large  and  a  second  correction 
gives  4 .  264430 .  The  same  root  found  by  Horner's  Method 
is  4.2644299.. 

Fourier's  conditions. 

Art.  106.  It  may  happen  that  after  several  corrections 
have  been  made  by  "Newton's  Method  it  will  be  found  that 
we  had  approached  the  true  value  of  the  root  for  a  while 
and  then  receded  from  it;  this  arises  from  over  correction. 
The  value  of  u,  the  correction,  was  obtained  approximately, 
as  will  be  remembered,  by  dropping  its  higher  powers,  and 


102  PRINCIPLES    OF    ALGEBRA. 

it  may  have  thus  happened  that  the  value  of  u  in  some 
case  was  too  large  and  when  added  to  the  approximate  root 
has  carried  the  value  beyond  the  truth;  and  if  this  be  again 
corrected  in  the  same  sense  the  new  result  will  be  still  fur- 
ther off;  and  would  be  an  approach  to  some  other  root. 

It  is  necessary  then  to  know,  not  only  that  there  is  a  root 
between  certain  limits,  but  also  that  there  is  no  other  root 
within  those  same  limits.  In  fact  it  has  been  demonstrated 
by  Fourier  that 

1 .  The  limits  between  which  the  required  root  exists 
must  be  so  narrow  as  to  contain  no  other  root  of  the  given 
equation ;  nor  yet  of  the  other  two  equations  obtained  by 
putting  the  first  and  second  derived  polynomials  equal  to 
zero. 

2.  That  the  approximation  must  start /rom  that  value 
which  makes  the  fir  st  member  and  its  second  derived  polynomial 
have  the  same  sign. 

But  it  is  not  deemed  necessary  to  give  the  demonstration 
of  these  principles  because  their  application  takes  away  from 
that  simplicity  and  expeditiousness  which  are  characteristic 
of  the  method  of  Newton.  By  that  method,  as  it  is  given 
above  and  as  it  was  left  by  its  immortal  author,  good,  prac- 
tical results  can  always  be  obtained.  If  at  any  time  it  is 
suspected  that  the  approximate  root  is  departing  from  the 
true  value  instead  of  approaching  it,  the  matter  may  be 
determined  at  once  by  substituting  in  the  proposed  equa- 
tion. 

If  the  student  is  desirous  of  taking  more  trouble  than  is 
involved  in  the  simple  application  of  Newton's  Method  it 
would  probably  be  better,  at  once  to  apply  Horner's 
Method. 

The  demonstration  of  Fourier's  Conditions  may  be  found 
in  Hackley's  Algebra,  Todhunter's  Theory  of  Equations 
and  elsewhere. 


TRIOONOMETKICAL  SOLUTION.  103 


CHAPTER  IX. 

Trigonometrical  Solution  of  Equations  of  the 
Third  Degree. 

Art.  107.  The  Irreducible  Case  of  these  equations  may 
be  solved  by  calling  in  the  aid  of  Trigonometry.  From 
that  branch  of  mathematics  we  know  that,  calling  w  any 
arc, 

cos2i^  =  2cos'^ti — 1, (1) 

Now,  cos3ft  =  cos(i6-l-2<t)  =  cosi6cos2it — sinM,sin2i6,  and 
this  is  equal  to  cositcos2M- — -sinM2siniicosit  =  cosw(cos2it — 
2sin'^i6) .  But  2sin'^i/.  =  1 — cos2it,  and  by  substitution  in 
the  last  expression,  we  have: 

cos3w.  =  2cos2ucosi^ — cosw., (2) 

Substitute  in  (2)  the  value  of  cos2m-  from  (1),  and  cos  3it  == 
4cos'^i6 — 3cosM-,  whence  we  derive: 

cos'w. — fcosu — Jcos3<^  =  0, (3) 

Now  suppose  that  we  had  a  cubic  equation  to  solve  which 
gave  rise  to  the  irreducible  case .  It  may  be  placed  in  the 
form 

^+P^+?  =  0,......(4) 

and  since  its  roots  are  all  real,  |^  >  J-,  and  p  is  negative. 

Let  X  =  rcosii,  wherein  r  is  a   constant  yet  to  be  de- 

termined.     Then   cosu  =    ,   and   substituting  in   (3)   we 

^      dx      1 
have  -^  —  -r- -7Cos3w,  =  0,  whence 

a^ ~x  —  -  cosdu  =  0, (5) 


104  PRINCIPLES  OF  ALGEBRA. 


By  comparing  (4)  and  (5),  we  see  that   —  =  ~p 
and  r  =  2^  -7.-  .     Also  that 


-  ^/^^• 


^  -  iiuu  /■  =  ^^1  -—^  .     Also  inat    — J —    =r  — q- 


-r  X  rcos3i^  =:  —q,  or  —^Xrco?,'^iL  =  —q:  .  ■ .  cos3i6=  —  = 
4:  o  pr 

/ p  / j>3.     Now,  since  q  and  ;?  are  known  and  p 

is  negative,  we  know  the  numerical  value  of  the  cosine  of 
du  and  having  found  Sit  from  the  Tables  of  Natural  Sines 
and  Cosines,  we  know  it  and  can  find  its  cosine .  This  be- 
ing multiplied  by  r  gives  the  value  of  x,  which  is  a  root  of 
the  proposed  equation.    The  numerical  value  of  ?•  comes  of 

course  from  r  =^  2^  /    P. 
\  3" 

Thus   we   have   one   of  the   real   roots;    but   the    value 


2    / — p^  =  not  only  cos3it,  but   also  cos(360°  — -  3w)  and 


■■^ 


27 

cos(360^  -f  3u),  consequently  the  cosines  of  the  thirds  of 
these  latter  two  arcs  will  be  the  remaining  roots,  after  hav- 
ing been  multiplied  by  r.' 

Art.  108.  Should  it  happen  that  3?^  =  180'  or  any  mul- 
tiple thereof,  cos(360°— 3i0  and  cos(360°-f 3^^)  would  be 
equal,  and  the  roots  corresponding  would  be  equal;  but 
they  might  have  been  discovered  and  divided  out  in  the 
first  instance,  when  no  resort  to  Cardan's  Formula  would 
have  been  necessary. 


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